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Homework Problems
Stat 490C
Chapter 2
1.
Model 1 is a uniform distribution from 0 to 100. Determine the table entries for a
generalized uniform distribution covering the range from a to b where a < b.
2.
Let X be a discrete random variable with probability function p(x) = 2(1/3)x for x
= 1, 2, 3, … What is the probability that X is odd?
3.
For a distribution where x > 2, you are given:


The hazard rate function: h(x) = z2/2x , for x > 2
F(5) = 0.84.
Calculate z.
4.
FX(t) = (t2-1)/9999 for 1<x<100. Calculate fX(50).
5.
You are given that the random variable X is distributed as a Weibull distribution
with parameters θ = 3 and τ = 0.5. Calculate:
a. Pr[X < 5]
b. Pr[3 < X < 5]
6.
(Spreadsheet Problem) You are given that the random variable X is distributed as
a Geometric distribution with parameters β = 3. Calculate:
a. Pr[X < 5]
b. Pr[3 < X < 5]
7.
A Weibull Distribution with parameter τ = 1 becomes what distribution?
8.
A random variable X has a density function f(x) = 4x(1+x2)-3, for x > 0.
Determine the mode of X.
Chapter 3
9.
Determine the following for a generalized uniform distribution covering the range
from a to b where a < d < b:
a. E[Xk]
b. E[X]
c. Var(X)
d. e(d)
10. For the Pareto distribution, determine E[X], Var(X), and the coefficient of
variation in terms of α and θ.
Homework Problems
Stat 490C
11. For the Gamma distribution, determine E[X], Var(X), the coefficient of variation,
and skewness in terms of α and θ.
12. For the Exponential distribution, determine E[X], Var(X), and e(d) in terms of θ.
13. You are given:
x3/27, for 0 < x < 3
F(x) =
1, for x >3
Calculate:
a.
b.
c.
d.
e.
f.
E[X]
Var(X)
e(1)
E[(X-1)+]
E[X Λ 2]
The Median
14. (Spreadsheet) If you roll two fair die, X is the sum of the dice. Calculate:
a.
b.
c.
d.
e.
f.
g.
E[X]
Var(X)
e(4)
E[(X-4)+]
E[X Λ 10]
The Mode
π20
15. You are given a sample of 2, 2, 3, 5, 8. For this empirical distribution, determine:
a. The mean
b. The variance
c. The standard deviation
d. The coefficient of variation
e. The skewness
f. The kurtosis
g. The probability generating function
16. The random variable X is distributed exponentially with θ = 0.2. Calculate
E[e2X].
17. * Losses follow a Pareto distribution has parameters of α = 7 and θ = 10,000.
Calculate e(5,000)
Homework Problems
Stat 490C
18. The amount of an individual claim has a Pareto distribution with θ = 8000 and α =
9. Use the central limit theorem to approximate the probability that the sum of
500 independent claims will exceed 550,000.
Chapter 4
19. The distribution function for losses from your renter’s insurance is the following:
F(x) = 1 – 0.8[1000/(1000+x)]5 – 0.2[12000/(12000+x)]3
Calculate:
a. E[X]
b. Var(X)
c. Use the normal approximation to determine the probability that the sum of
100 independent claims will not exceed 200,000.
20. * X has a Burr distribution with parameters α = 1, γ = 2, and θ = 10000.5. Y has a
Pareto distribution with parameters α = 1 and θ = 1000. Z is a mixture of X and Y
with equal weights on each component. Determine the median of Z.
21. The random variable X is distributed as a Pareto distribution with parameters α
and θ. E(X) = 1 and Var(X) = 3. The random variable Y is equal to 2X.
Calculate the Var(Y).
22. * Claim severities are modeled using a continuous distribution and inflation
impacts claims uniformly at an annual rate of s. Which of the following are true
statements regarding the distribution of claim severities after the effect of
inflation?
i. An exponential distribution will have a scale parameter of (1+s)θ.
ii. A Pareto distribution will have scale parameters (1+s)α and (1+s)θ.
iii. An Inverse Gaussian distribution will have a scale parameter of
(1+s)θ.
23. * The aggregate losses of Eiffel Auto Insurance are denoted in euro currency and
follow a Lognormal distribution with μ = 8 and σ = 2. Given that 1 euro = 1.3
dollars, determine the lognormal parameters for the distribution of Eiffel’s losses
in dollars.
24. (Spreadsheet) Calculate Γ(1.5).
25. The random variable X is the number of dental claims in a year and is distributed
as a gamma distribution given parameter θ and with parameter α = 1. θ is
distributed uniformly between 1 and 3. Calculate E(X) and Var(X).
Homework Problems
Stat 490C
26. A dental insurance company has 1000 insureds. Assume the number of claims
from each insured is independent. Using the information in Problem 25 and the
normal approximation, calculate the probability that the Company will incur more
than 2100 claims.
27. * Let N have a Poisson distribution with mean Λ. Let Λ have a uniform
distribution on the interval (0,5). Determine the unconditional probability that
N > 2.
28. The number of hospitalization claims in a year follows a Poisson distribution with
a mean of λ. The probability of exactly three claims during a year is 60% of the
probability that there will be 2 claims. Determine the probability that there will
be 5 claims.
29. If the number of claims is distributed as a Poison distribution with λ = 3,
calculate:
a. Pr(N = 0)
b. Pr(N = 1)
c. Pr(N = 2)
d. E(N)
e. Var(N)
30. If the number of claims is distributed as a zero truncated Poison distribution with
λ = 3, calculate:
a. Pr(N = 0)
b. Pr(N = 1)
c. Pr(N = 2)
d. E(N)
e. Var(N)
31. If the number of claims is distributed as a zero modified Poison distribution with
λ = 3 and p0M = 0.5, calculate:
a. Pr(N = 0)
b. Pr(N = 1)
c. Pr(N = 2)
d. E(N)
e. Var(N)
32. If the number of claims is distributed as a Geometric distribution with β = 3,
calculate:
a. Pr(N = 0)
b. Pr(N = 1)
c. Pr(N = 2)
d. E(N)
e. Var(N)
Homework Problems
Stat 490C
33. (Spreadsheet) If the number of claims is distributed as a Binomial distribution
with m = 6 and q = 0.4, calculate:
a. Pr(N = 0)
b. Pr(N = 1)
c. Pr(N = 2)
d. E(N)
e. Var(N)
34. (Spreadsheet) If the number of claims is distributed as a Negative Binomial
distribution with γ = 3 and β = 2, calculate:
a. Pr(N = 0)
b. Pr(N = 1)
c. Pr(N = 2)
d. E(N)
e. Var(N)
35. (Spreadsheet) If the number of claims is distributed as a Logarithmic distribution
with β = 3, calculate:
a. Pr(N = 0)
b. Pr(N = 1)
c. Pr(N = 2)
d. E(N)
e. Var(N)
36. * The Independent Insurance Company insures 25 risks, each with a 4%
probability of loss. The probabilities of loss are independent. What is the
probability of 4 or more losses in the same year? (Hint: Use the binomial
distribution.)
37. * You are given a negative binomial distribution with γ = 2.5 and β = 5. For what
value of k does pk take on its largest value?
38. * N is a discrete random variable form the (a, b, 0) class of distributions. The
following information is known about the distribution:
 P(N = 0) = 0.327680
 P(N = 1) = 0.327680
 P(N = 2) = 0.196608
 E[N] = 1.25
Based on this information, which of the following are true statements?
I.
II.
III.
P(N = 3) = 0.107965
N is from a binomial distribution.
N is from a Negative Binomial Distribution.
Homework Problems
Stat 490C
39. * You are given:
 Claims are reported at a Poisson rate of 5 per year.
 The probability that a claim will settle for less than 100,000 is 0.9.
What is the probability that no claim of 100,000 or more will be reported in the
next three years?
40. N is distributed as a zero modified geometric distribution with β = 3 and p0M =
0.5. Calculate:
a. e(2)
b. Var[N ^ 3]
c. E[(N-2)+]
Chapter 5
41. Losses are distributed Exponentially with θ = 1000. Losses are subject to an
ordinary deductible of 500. Calculate:
a. fYL(y)
b. FYL(y)
c. SYL(y)
d. E(YL)
e. The loss elimination ratio
42. Losses are distributed Exponentially with parameter θ = 1000. Losses are subject
to an ordinary deductible of 500. Calculate:
a. fYP(y)
b. FYP(y)
c. E(YP)
43. Losses are distributed Exponentially with θ = 1000. Losses are subject to a
franchise deductible of 500. Calculate:
a. fYL(y)
b. FYL(y)
c. SYL(y)
d. E(YL)
e. The loss elimination ratio
44. Losses are distributed Exponentially with parameter θ = 1000. Losses are subject
to a franchise deductible of 500. Calculate:
a. fYP(y)
b. FYP(y)
c. E(YP)
Homework Problems
Stat 490C
45. Last year, losses were distributed Exponentially with θ = 1000. This year losses
are subject to 10% inflation. Losses in both years are subject to an ordinary
deductible of 500. Calculate the following for this year:
a. E(YL)
b. E(YP)
46. * Losses follow an exponential distribution with parameter θ. For a deductible of
100, the expected payment per loss is 2000. What is the expected payment per
loss in terms of θ for a deductible of 500?
47. * In year 2007, claim amounts have the following Pareto distribution
F(x) = 1 – (800/[x+800])3
The annual inflation rate is 8%. A franchise deductible of 300 will be
implemented in 2008. Calculate the loss elimination ration of the franchise
deductible.
48. (Spreadsheet) If you roll two fair die, X is the sum of the dice. Let X represent
the distribution of losses from a particular event. If an ordinary deductible of 3 is
applied, calculate:
a.
b.
c.
d.
E(YL)
E(YP)
Var(YL)
Var(YP)
49. Losses are distributed uniformly between 0 and 100,000. An insurance policy
which covers the losses has a 10,000 deductible and an 80,000 upper limit. The
upper limit is applied prior to applying the deductible. Calculate:
a.
b.
c.
d.
E(YL)
E(YP)
Var(YL)
Var(YP)
50. Last year, losses were distributed Exponentially with θ = 1000. This year losses
are subject to 25% inflation. Losses in both years are subject to an ordinary
deductible of 500, an upper limit of 4000, and coinsurance of 20% Calculate the
following for this year:
a. E(YL)
b. E(YP)
Homework Problems
Stat 490C
51. * An insurance company offers two types of policies: Type Q and Type R. Type
Q has no deductible, but has a policy limit of 3000. Type R has no limit, but has
an ordinary deductible of d. Losses follow a Pareto distribution with θ = 2000
and α = 3.
Calculate the deductible d such that both policies have the same expected cost per
loss.
52. * Well Traveled Insurance Company sells a travel insurance policy that
reimburses travelers for any expense incurred for a planned vacation that is
cancelled because of airline bankruptcies. Individual claims follow a Pareto
distribution with α = 2 and θ = 500. Well Traveled imposes a limit of 1000 on
each claim. If a planned policyholder’s planned vacation is cancelled due to
airline bankruptcy and he or she has incurred more than 1000 of expenses, what is
the expected non-reimbursable amount of the claim?
53. If X is uniformly distributed on from 0 to b. An ordinary deductible of d is
applied. Calculate:
a. E[YL]
b. Var[YL]
c. E[YP]
d. Var[YP]
54. Losses follow a Pareto distribution with α = 3 and θ = 200. A policy covers the
loss except for a franchise deductible of 50. X is the random variable
representing the amount paid by the insurance company per payment. Calculate
the expected value of the X.
55. Losses in 2007 are uniformly distributed between 0 and 100,000. An insurance
policy pays for all claims in excess of an ordinary deductible of $20,000. For
2008, claims will be subject to uniform inflation of 20%. The Company
implements an upper limit u (without changing the deductible) so that the
expected cost per loss in 2008 is the same as the expected cost per loss in 2007.
Calculate u.
56. Losses follow an exponential distribution with a standard deviation of 100. An
insurance company applies a policy deductible d which results in a Loss
Elimination Ratio of 0.1813. Calculate d.
57. Losses are distributed following the single parameter Pareto (Not the same as the
Pareto distribution in Question 54) with α = 4 and θ = 90. An insurance policy is
issued with a deductible of 100. Calculate the Var(YL).
Homework Problems
Stat 490C
58. Losses follow a Pareto distribution with α = 5 and θ = 2000. A policy pays 100%
of losses from 500 to 1000. In other words, if a loss occurs for less than 500, no
payment is made. If a loss occurs which exceeds 1000, no payment is made.
However, if a loss occurs which is between 500 and 1000, then the entire amount
of the loss is paid. Calculate E(YP).
59. * You are given the following:
 Losses follow a lognormal distribution with parameters μ =10 and σ = 1.
 For each loss less than or equal to 50,000, the insurer makes no payment.
 For each loss greater than 50,000, the insurer pays the entire amount of the
loss up to the maximum covered loss of 100,000.
Determine the expected amount of the loss.
60. You are given the following:
 Losses follow a distribution prior to the application of any deductible with
a mean of 2000.
 The loss elimination ratio at a deductible of 1000 is 0.3.
 60% of the losses in number are less than the deductible of 1000.
Determine the average size of the loss that is less than the deductible of 1000.
61. Losses follow a Pareto distribution with α = 5 and θ = 2000. An insurance policy
covering these losses has a deductible of 100 and makes payments directly to the
physician. Additionally, the physician is entitled to a bonus if the loss is less than
500. The bonus is 10% of the difference between 500 and the amount of the loss.
The following table should help clarify the arrangement:
Amount of Loss
50
100
250
400
500
1000
Loss Payment
0
0
150
300
400
900
Bonus
45
40
25
10
0
0
Calculate the expected total payment (loss payment plus bonus) from the
insurance policy to the physician per loss.
Homework Problems
Stat 490C
Chapter 6
62. * For an insured, Y is the random variable representing the total amount of time
spent in a hospital each year. The distribution of the number of hospital
admissions in a year is:
Number of Admissions
0
1
2
Probability
0.60
0.30
0.10
The distribution of the length of each stay for an admission follows a Gamma
distribution with α = 1 and θ = 5.
Calculate E[Y] and Var[Y].
63. For an insurance company, each loss has a mean of 100 and a variance of 100.
The number of losses follows a Poisson distribution with a mean of 500. Each
loss and the number of losses are mutually independent.
The loss ratio for the insurance company is defined as the ratio of aggregate losses
to the total premium collected.
The premium collected is 110% of the expected aggregate losses.
Using the normal approximation, calculate the probability that the loss ratio will
exceed 95%.
64. An automobile insurer has 1000 cars covered during 2007. The number of
automobile claims for each car follows a negative binomial distribution with β = 1
and γ = 1.5. Each claim is distributed exponentially with a mean of 5000.
Assume that the number of claims and the amount of the loss are independent and
identically distributed.
Using the normal distribution as an approximating distribution of aggregate
losses, calculate the probability that losses will exceed 8 million.
Homework Problems
Stat 490C
65. X1, X2, and X3 are mutually independent random variables. These random
variables have the following probability functions:
x
0
1
2
3
f1(x)
0.6
0.4
0.0
0.0
f2(x)
0.4
0.3
0.2
0.1
f3(x)
0.1
0.4
0.5
0.0
Calculate fS(x).
66. The number of claims follows a Poisson distribution with a mean of 3. The
distribution of claims is fX(1) = 1/3 and fX(2) = 2/3. Calculate fS(4).
67. (Spreadsheet) The number of claims and the amount of each claim is mutually
independent. The frequency distribution is:
n
1
2
3
4
5
6
pn
0.05
0.20
0.30
0.25
0.15
0.05
The severity distribution is:
x
100
200
300
400
500
600
700
800
fX(x)
0.25
0.20
0.15
0.12
0.10
0.08
0.06
0.04
Calculate fS(x). (See Example 6.6 in the book for an example.)
Homework Problems
Stat 490C
68. You are given the following table for aggregate claims:
S
0
100
200
300
400
500
600
FS(s)
E[(X-d)+]
0.50
0.65
0.75
60
Losses can only occur in multiples of 100. Calculate the net stop loss premium
for stop loss insurance covering losses in excess of 325.
69. * Losses follow a Poisson frequency distribution with a mean of 2 per year. The
amount of a loss is 1, 2, or 3 with each having a probability of 1/3. Loss amounts
are independent of the number of losses and from each other.
An insurance policy covers all losses in a year subject to an annual aggregate
deductible of 2. Calculate the expected claim payments for this insurance policy.
70. Purdue University has decided to provide a new benefit to each class at the
university. Each class will be provided with group life insurance. Students in
each class will have 10,000 of coverage while the professor for the class will have
20,000. STAT 490C has 27 students all age 22 split 12 males and 15 females.
The class also has an aging professor. The probability of death for each is listed
below:
Age
Gender
Probability of Death
22
Male
.005
22
Female
.003
52
Male
.020
Purdue purchases a Stop Loss Policy such with an aggregate deductible of 20,000.
Calculate the net premium that Purdue will pay for the stop loss coverage.
Homework Problems
Stat 490C
71. For an automobile policy, the severity distribution is Gamma with α = 3 and θ =
1000. The number of claims in a year is distributed as follows:
Number of Claims
1
2
3
4
Probability
.65
.20
.10
.05
Calculate:
a. E(S)
b. Var(S)
c. MS(t)
72. (Spreadsheet) You are given that the number of losses follow a Poisson with λ =
6. Losses are distributed as follows: fX(1) = fX(2) = fX(4) = 1/3. Find fS(x) for
total losses of 100 or less.
73. On a given day, a doctor provides medical care for NA adults and NC children.
Assume that NA and NC have Poisson distributions with parameters 3 and 2
respectively. The distributions for the length of care per patient are as follows:
Time
1 hour
2 hours
Adult
0.4
0.6
Child
0.9
0.1
Let NA, NC, and the lengths of stay for all patients be independent. The doctor
charges 200 per hour. Determine the probability that the office income on a given
day will be less than or equal to 800.
74. (Spreadsheet) Losses follow a Pareto distribution with α = 1 and θ = 1000.
a. Using the Method of Rounding with a span of 100, calculate fj for 0 < j
<1000. Calculate the mean of the discretized distribution. Compare this
mean to the actual mean.
b. Using the Method of Local Moment Matching and matching the mean
(with k = 1) and a span of 100, calculate fj for 0 < j <1000. Calculate the
mean of the discretized distribution. Why does this mean not equal the
actual mean?
75. Using the example that was handed out in class, find fS(6) and fS(7).
76. * You are given:
a. S has a compound Poisson distribution with λ = 2; and
b. Individual claim amounts x are distributed as follows:
fX(1) = 0.4 and fX(2) = 0.6
Determine fS(4).
Homework Problems
Stat 490C
77. * Aggregate claims S has a compound Poisson distribution with discrete
individual claim amount distributions of fX(1) = 1/3 and fX(3) = 2/3.
Also, fS(4) = fS(3) + 6fS(1).
Calculate the Var(S).
78. * For aggregate claims S, you are given:

a.
50
fS(X) =  f ( x)  e
n 0
*n
(50)
X
n!
n
b. Losses are distributed as follows:
x
1
2
3
fX(x)
0.4
0.5
0.1
Determine Var(S)
79. With no deductible, the number of payments for losses under warranty coverage
for an Iphone follows a negative binomial distribution with a mean of 0.25 and a
variance of 0.375.
The company imposes a deductible of d such that the number of expected
payments for losses is reduced to 50% of the prior expected payments.
Calculate the Var (NP) after the imposition of the deductible.
80. * Prior to the application of any deductible, aggregate claim counts during 2005
followed a Poisson with λ = 14. Similarly, individual claim sizes followed a
Pareto with α = 3 and θ = 1000.
Annual inflation for the claim sizes is 10%.
All policies in 2005 and 2006 are subject to a 250 ordinary deductible.
Calculate the increase in the number of claims that exceed the deductible in 2006
when compared to 2005.
Homework Problems
Stat 490C
81. * An insurance portfolio produces N claims with the following distribution:
n
0
1
2
P(N=n)
0.1
0.5
0.4
Individual claim amounts have the following distribution:
x
0
10
20
fX(x)
0.7
0.2
0.1
Individual claim amounts and counts are independent.
Stop Loss insurance is purchased with an aggregate deductible of 400% of
expected claims.
Calculate the net stop loss premium.
82. Losses are modeled assuming that the amount of all losses is 40 and that the
number of losses follows a geometric distribution with a mean of 4. Calculate the
net stop loss premium for coverage with an aggregate deductible of 100.
83. * An aggregate claim distribution has the following characteristics:
P[S = i] = 1/6 for i = 1, 2, 3, 4, 5, or 6. A stop loss insurance with a deductible
amount of d has an expected insurance payment of 1.5
Determine d.
84. Purdue University has decided to provide a new benefit to each class at the
university. Each class will be provided with group life insurance. Students in
each class will have 10,000 of coverage while the professor for the class will have
20,000. STAT 490C has 27 students all age 22 split 12 males and 15 females.
The class also has an aging professor. The probability of death for each is listed
below:
Age
Gender
Probability of Death
22
Male
.005
22
Female
.003
52
Male
.020
The insurance company providing the coverage charges a premium equal to the
expected claims plus one standard deviation. Calculate the premium.
Homework Problems
Stat 490C
85. * An insurer provides life insurance for the following group of independent lives:
Number of Lives
100
200
300
Death Benefit
1
2
3
Probability of Death
0.01
0.02
0.03
The insurer purchases reinsurance with a retention of 2 on each life.
The reinsurer charges a premium of H equal to the its expected claims plus the
standard deviation of its claims.
The insurer charges a premium of G which is equal to its expected retained claims
plus the standard deviation of the retained claims plus H
Calculate G.
86. * Two portfolios of independent insurance policies have the following
characteristics:
Number in
Class
2000
500
Portfolio A
Probability of
Claim
0.05
0.10
Class
Number in
Class
Portfolio B
Probability of
Claim
1
2
2000
500
0.05
0.10
Class
1
2
Claim Amount
Per Policy
1
2
Claim Amount
Distribution
Mean Variance
1
1
2
4
The aggregate claims in the portfolios are denoted by SA and SB, respectively.
Calculate Var[SB]/Var[SA].
Homework Problems
Stat 490C
87. * An insurance company is selling policies to individuals with independent future
lifetimes and identical mortality profiles. For each individual, the probability of
death by all causes is 0.10 and the probability of death due to an accident is 0.01.
Each insurance policy pays a benefit of 10 for an accidental death and 1 for nonaccidental death.
The company wishes to have at least 95% confidence that premiums with a
relative security loading of 0.20 are adequate to cover claims. (In other words,
the premium is 1.20E(S).)
Using the normal approximation, determine the minimum number of policies that
must be sold.
88. Chen Casualty Company writes insurance the covers tsunamis only. If a tsunami
occurs, Chen pays a benefit of 1 million. The benefit is paid at the time of the
tsunami. The distribution of annual tsunami claims is as follows:
Number of Tsunamis
0
1
2
Probability
0.90
0.08
0.02
Chen collects a premium of 0.15 million at the beginning of the year.
Chen does not have any expenses nor does the Company earn interest.
Chen has a surplus equal to ten times one year’s expected claims.
Calculate the probability that Chen is still solvent after 2 years.
89. Brown Assurance Company has a surplus of 3 at the end of 2006. Each year
Brown collects premiums of 5 at the beginning of the calendar year. During the
calendar year, Brown earns 10% interest. At the end of the calendar uear, Brown
pays any losses incurred. The distribution of annual losses is 0, 5, and 10 with a
probability of 0.3, 0.5, and 0.2 respectively.
Calculate the probability of ruin prior to the end of 2009.
Homework Problems
Stat 490C
90. Bauer Blasting Company (BBC) self insures its employee life insurance. At the
end of 2006, the life insurance fund has a surplus of 5,000. At the beginning of
each calendar year, BBC contributes 20,000 into the life insurance fund. The
fund earns interest at 4%.
If a BBC employee dies on the job, BBC provides a death benefit of $10,000 per
death at the end of the calendar year. The distribution of deaths each year is:
Deaths
1
2
3
Probability
0.2
0.5
0.3
Calculate the probability of ruin for the life insurance fund within 3 years.
91. * You are given:
a. Initial Surplus is 10.
b. Annual losses are distributed as follows:
Annual Losses
0
10
20
30
Probability
0.60
0.25
0.10
0.05
c. Premiums paid at the beginning of the year are equal to expected losses
for the year.
d. If surplus increases in the year, a dividend is paid at the end of the year.
The dividend is equal to half the increase in surplus.
e. There are no other cash flows.
Calculate the probability of ruin during the first two years.
92. * The number of major hurricanes that hit the island nation of Justcoast is given
by a Poisson process with 0.100 storms expected per year.
Justcoast establishes a fund that will pay 100 per storm. The fund charges an
annual premium, payable at the start of each year, of 10. At the start of this year
(before the premium is paid), the fund has 65.
Claims are paid immediately when there is a storm. If the fund ever runs out of
money, it immediately ceases to exist.
There are no expenses and the fund does not earn any interest.
What is the probability that the fund is still functioning in 10 years?
Homework Problems
Stat 490C
93. * A random sample, X1, X2, …, Xn, is drawn from a distribution with a mean of
2/3 and a variance of 1/18.
ˆ = (X1 + X2 + … + Xn)/(n-1) is the estimator of the distribution mean θ.
Find MSE( ˆ ).
94. * Claim sizes are uniformly distributed over the interval [0, θ]. A sample of 10
claims, denoted by X1, X2, …,X10 was observed and an estimate of θ was obtained
using:
ˆ = Y = max(X1, X2, …,X10)
Recall that the probability density function for Y is:
fY(y) = 10y9/θ10
Calculate the mean square error for ˆ for θ =100.
95. * You are given two independent estimates of an unknown quantity θ:
a. Estimator A: E( ˆ A) = 1000 and σ( ˆ A) = 400
b. Estimator B: E( ˆ B) = 1200 and σ( ˆ B) = 200
Estimator C is a weighted average of Estimator A and Estimator B such that:
ˆ C = (w) ˆ A + (1-w) ˆ B
Determine the value of w that minimizes σ( ˆ C).
Homework Problems
Stat 490C
96. * You are given:
x
0
1
2
3
Pr(X = x)
0.5
0.3
0.1
0.1
Using a sample size of n, the population mean is estimated by the sample
mean X . The variance is estimated by:
S
2
n
=
(X
i
 X)
2
n
Calculate the bias of S2n when n = 4.
97. * For the random variable X, you are given:
a. E(X) = θ, θ > 0
b. Var(θ) = θ2/25
c. ˆ = kX/(k+1)
d. MSEθ(θ) = 2[bias θ(θ)]2
Determine k.
98. You are given the following sample of claims:
X: 12, 13, 16, 16, 22, 24, 26, 26, 28, 30
The sum of X is 213 and the sum of X2 is 4921.
H0 is that μx = 17 and H1 is that μx > 17.
Calculate the z statistic, the critical value(s) assuming a significance level of 1%,
and the p value. State your conclusion with regard to the Hypothesis Testing.
Homework Problems
Stat 490C
Wang Warranty Corporation is testing iPods. Wang starts with 100 iPods and tests
them by dropping them on the ground. Wang records the number of drops before each
iPod will no longer play. The following data is collected from this test:
Drops to Failure
1
2
4
6
7
8
Number
6
2
3
4
4
5
Drops to Failure
9
10
11
12
13
22
Number
7
6
7
7
48
1
Ledbetter Life Insurance Company is completing a mortality study on a 3 year term
insurance policy. The following data is available:
Life
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Date of Entry
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.5
0.5
1.0
1.0
2.0
Date of Exit
0.2
0.3
0.4
0.5
0.5
0.5
1.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
2.0
1.0
3.0
3.0
2.5
Reason for Exit
Lapse
Death
Lapse
Death
Death
Lapse
Death
Expiry of Policy
Expiry of Policy
Expiry of Policy
Expiry of Policy
Expiry of Policy
Expiry of Policy
Expiry of Policy
Expiry of Policy
Lapse
Death
Expiry of Policy
Expiry of Policy
Death
Schneider Trucking Company had the following losses during 2006:
Amount
of Claim
0 – 10
10-20
20-30
30 +
Number
of Payments
8
5
4
3
Total Amount
of Losses
60
70
110
200
Total
20
440
Homework Problems
Stat 490C
99. Using the data from Wang Warranty Corporation, calculate:
a. p100(x)
b. F100(x)
c. The empirical mean
d. The empirical variance
e. Ĥ(x) where Ĥ(x) is the cumulative hazard function from the Nelson Åalen
estimate
f. Ŝ(x) where Ŝ(x) is the survival function from the Nelson Åalen estimate
100. Using the data from Schneider Trucking Company, calculate:
a. The ogive, F20(x)
b. The histogram, f20(x)
c. E(XΛ30) minus E[(X-20)+]
101. * You are given:
Claim Size (X)
(0,25]
(25,50]
(50,100]
(100,200]
Number of Claims
30
32
20
8
Assume a uniform distribution of claim sizes within each interval.
Estimate the mean of the claim size distribution.
Estimate the second raw moment of the claim size distribution.
Chapter 11
102. A mortality study is completed on 30 people. The following deaths occur during
the five years:
3 deaths at time 1.0
4 deaths at time 2.0
5 deaths at time 3.0
8 deaths at time 3.8
10 deaths at time 4.5
There were no other terminations and no lives entered the study after the start of the
study.
The data was smoothed using a uniform kernel with a bandwidth of 1. Calculate fˆ (x)
and F̂ (x) for all x > 0.
Homework Problems
Stat 490C
103. * From a population having a distribution function F, you are given the following
sample:
2.0, 3.3, 3.3, 4.0, 4.0, 4.7, 4.7, 4.7
Calculate the kernel density estimate of F(4), using a uniform kernel with
bandwidth of 1.4.
104. * You study five lives to estimate the time from the onset of a disease until death.
The times to death are:
2 3 3 3 7
Using a triangular kernel with a bandwidth of 2, estimate the density function at
2.5.
105. You are given the following random sample:
12
15
27
42
The data is smoothed using a uniform kernel with a bandwidth of 6.
Calculate the mean and variance of the smoothed distribution.
106. You are given the following random sample:
12
15
27
42
The data is smoothed using a triangular kernel with a bandwidth of 12.
Calculate the mean and variance of the smoothed distribution.
107. You are given the following random sample:
12
15
27
42
The data is smoothed using a gamma kernel with a bandwidth of 3.
Calculate the mean and variance of the smoothed distribution.
Homework Problems
Stat 490C
Chapter 12
108. You are given the following sample of claims obtained from an inverse gamma
distribution:
X: 12, 13, 16, 16, 22, 24, 26, 26, 28, 30
The sum of X is 213 and the sum of X2 is 4921.
Calculate α and θ using the method of moments.
109. * You are given the following sample of five claims:
4 5 21 99 421
Find the parameters of a Pareto distribution using the method of moments.
110. * A random sample of death records yields the follow exact ages at death:
30 50 60 60 70 90
The age at death from which the sample is drawn follows a gamma distribution.
The parameters are estimated using the method of moments.
Determine the estimate of α.
111. * You are given the following:

The random variable X has the density function
f(x) = αx-α-1 , 1 < x < ∞, α >1

A random sample is taken of the random variable X.
Calculate the estimate of α in terms of the sample mean using the method of
moments.
112. * You are given the following:
 The random variable X has the density function f(x) = {2(θ – x)}/θ2, 0 < x < θ
 A random sample of two observations of X yields values of 0.50 and 0.70.
Determine θ using the method of moments.
Homework Problems
Stat 490C
113. You are given the following sample of claims:
X: 12, 13, 16, 16, 22, 24, 26, 26, 28, 30
Calculate the smoothed empirical estimate of the 40th percentile of this
distribution.
114. * For a complete study of five lives, you are given:
a. Deaths occur at times t = 2, 3, 3, 5, 7.
b. The underlying survival distribution S(t) = 4-λt, t > 0
Using percentile matching at the median, calculate the estimate of λ.
115. You are given the following 9 claims:
X: 10, 60, 80, 120, 150, 170, 190, 230, 250
The sum of X = 1260 and the sum of X2 = 227,400.
The data is modeled using an exponential distribution with parameters
estimated using the percentile matching method.
Calculate θ based on the empirical value of 120.
116. * For a sample of 10 claims, x1 < x2 < … < x10 you are given:
a. The smoothed empirical estimate of the 55th percentile is 380.
b. The smoothed empirical estimate of the 60th percentile is 402.
Determine x6.
117. You are given the following:
a. Losses follow a Pareto distribution with parameters α and θ.
b. The 10th percentile of the distribution is θ – k, where k is a constant.
c. The 90th percentile of the distribution is 5θ – 3k.
Determine α.
Homework Problems
Stat 490C
118. You are given the following random sample of 3 data points from a population
with a Pareto distribution with θ = 70:
X:
15
27
43
Calculate the maximum likelihood estimate for α.
119. * You are given:
a. Losses follow an exponential distribution with mean θ.
b. A random sample of 20 losses is distributed as follows:
Range
[0,1000]
(1000, 2000)
(2000,∞)
Frequency
7
6
7
Calculate the maximum likelihood estimate of θ.
120. * You are given the following:
 The random variable X has the density function f(x) = {2(θ – x)}/θ2, 0 < x < θ
 A random sample of two observations of X yields values of 0.50 and 0.90.
Determine the maximum likelihood estimate for θ.
121. * You are given:
a. Ten lives are subject to the survival function S(t) = (1-t/k)0.5, 0 < t < k
b. The first two deaths in the sample occured at time t = 10.
c. The study ends at time t = 10.
Calculate the maximum likelihood estimate of k.
122. * You are given the following:
a. The random variable X follows the exponential distribution with
parameter θ.
b. A random sample of three observations of X yields values of 0.30, 0.55,
and 0.80
Determine the maximum likelihood estimate of θ.
Homework Problems
Stat 490C
123. * Ten laboratory mice are observed for a period of five days. Seven mice die
during the observation period, with the following distribution of deaths:
Time of Death in Days
2
3
4
5
Number of Deaths
1
2
1
3
The lives in the study are subject to an exponential survival function with mean of
θ.
Calculate the maximum likelihood estimate of θ.
124. * A policy has an ordinary deductible of 100 and a policy limit of 1000. You
observe the following 10 payments:
15 50 170 216 400 620 750 900 900 900
An exponential distribution is fitted to the ground up distribution function, using
the maximum likelihood estimate.
Determine the estimated parameter θ.
125. * Four lives are observed from time t = 0 until death. Deaths occur at t = 1, 2, 3,
and 4. The lives are assumed to follow a Weibull distribution with τ = 2.
Determine the maximum likelihood estimator for θ.
126. * The random variable X has a uniform distribution on the interval [0,θ]. A
random sample of three observations of X are recorded and grouped as follows:
Interval
[0,k)
[k,5)
[5,θ]
Number of
Observations
1
1
1
Calculate the maximum likelihood estimate of θ
Homework Problems
Stat 490C
127. * A random sample of three claims from a dental insurance plan is given below:
225
525
950
Claims are assumed to follow a Pareto distribution with parameters θ = 150 and α.
Determine the maximum likelihood estimate of α.
128. * The following claim sizes are experienced on an insurance coverages:
100
500
1,000 5,000 10,000
You fit a lognormal distribution to this experience using maximum likelihood.
Determine the resulting estimate of σ.
You are given the following data from a sample:
k
0
1
2
3
4
5
nk
20
25
30
15
8
2
Use this data for the next four problems.
129. Assuming a Binomial Distribution, estimate m and q using the Method of
Moments.
130. Assuming a Binomial Distribution, find the MLE of q given that m = 6.
131. (Spreadsheet) Assuming a Binomial Distribution, find the MLE of m and q.
132. Assuming a Poisson Distribution, approximate the 90% confidence interval for
the true value of λ.
Homework Problems
Stat 490C
133. You are given the following 20 claims:
X: 10, 40, 60, 65, 75, 80, 120, 150, 170, 190, 230, 340, 430, 440, 980, 600,
675, 950, 1250, 1700
The data is being modeled using an exponential distribution with θ = 427.5.
Calculate D(200).
134. You are given the following 20 claims:
X: 10, 40, 60, 65, 75, 80, 120, 150, 170, 190, 230, 340, 430, 440, 980, 600,
675, 950, 1250, 1700
The data is being modeled using an exponential distribution with θ = 427.5.
You are developing a p-p plot for this data. What are the coordinates for x7 =
120.
135. Mark the following statements True or False with regard to the KolmogorovSmirnov test:
The Kolmogorov-Smirnov test may be used on grouped data as well as
individual data.
If the parameters of the distribution being tested are estimated, the critical
values do not need to be adjusted.
If the upper limit is less than ∞, the critical values need to be larger.
136. Balog’s Bakery has workers’ compensation claims during a month of:
100, 350, 550, 1000
Balog’s owner, a retired actuary, believes that the claims are distributed
exponentially with θ = 500.
He decides to test his hypothesis at a 10% significance level.
Calculate the Kolmogorov-Smirnov test statistic.
State the critical value for his test and state his conclusion.
He also tests his hypothesis using the Anderson-Darling test statistic. State the
values of this test statistic under which Mr. Schmidt would reject his hypothesis.
Homework Problems
Stat 490C
137. * The observations of 1.7, 1.6, 1.6, and 1.9 are taken from a random sample. You
wish to test the goodness of fit of a distribution with probability density function
given by f(x) = 0.5x for 0 < x < 2.
Using the Kolmogorov-Smirnov statistic, which of the following should you do?
a.
b.
c.
d.
e.
Accept at both levels
Accept at the 0.01 level but reject at the 0.10 level
Accept at the 0.10 level but reject at the 0.01 level
Reject at both levels
Cannot be determined.
138. * Two lives are observed beginning at time t=0. One dies at time 5 and the other
dies at time 9. The survival function S(t) = 1 – (t/10) is hypothesized.
Calculate the Kolmogorov-Smirnov statistic.
139. * From a laboratory study of nine lives, you are given:
a. The times of death are 1, 2, 4, 5, 5, 7, 8, 9, 9
b. It has been hypothesized that the underlying distribution is uniform
with ω = 11.
Calculate the Kolmogorov-Smirnov statistic for the hypothesis.
140. You are given the following data:
Claim Range
0-100
100-200
200-500
500-1000
1000+
Count
30
25
20
15
10
H0: The data is from a Pareto distribution.
H1: The data is not from a Pareto distribution.
Your boss has used the data to estimate the parameters as α = 4 and θ = 1200.
Calculate the chi-square test statistic.
Calculate the critical value at a 10% significance level.
State whether you would reject the Pareto at a 10% significance level.
Homework Problems
Stat 490C
141. During a one-year period, the number of accidents per day in the parking lot of
the Steenman Steel Factory is distributed:
Number of Accidents
0
1
2
3
4+
Days
220
100
30
10
5
H0: The distribution of the number of accidents is distributed as Poison with a
mean of 0.625.
H1: The distribution of the number of accidents is not distributed as Poison with a
mean of 0.625.
Calculate the chi-square statistic.
Calculate the critical value at a 10% significance level.
State whether you would reject the H0 at a 10% significance level.
142. * You are given the following random sample of automobile claims:
54
140
230
560
600 1,100 1,500 1,800 1,920 2,000
2,450 2,500 2,580 2,910 3,800 3,800 3,810 3,870 4,000 4,800
7,200 7,390 11,750 12,000 15,000 25,000 30,000 32,200 35,000 55,000
You test the hypothesis that automobile claims follow a continuous distribution
F(x) with the following percentiles:
x
F(x)
310
0.16
500
0.27
2,498
0.55
4,876
0.81
7,498
0.90
12,930
0.95
You group the data using the largest number of groups such that the expected
number of claims in each group is at least 5.
Calculate the Chi-Square goodness-of-fit statistic.
Homework Problems
Stat 490C
143. Based on a random sample, you are testing the following hypothesis:
H0: The data is from a population distributed binomial with m = 6 and q = 0.3.
H1: The data is from a population distributed binomial.
You are also given:
L(θ0) = .1 and L(θ1) = .3
Calculate the test statistic for the Likelihood Ratio Test
State the critical value at the 10% significance level
144. State whether the following are true or false

The principle of parsimony states that a more complex model is better
because it will always match the data better.

In judgment-based approaches to determining a model, a modeler’s
experience is critical.

In most cases, judgment is required in using a score-based approach to
selecting a model.
Chapter 17
145. A random number generated from a uniform distribution on (0, 1) is 0.6. Using
the inverse transformation method, calculate the simulated value of X assuming:

X is distributed Pareto with α = 3 and θ = 2000

F(x) =

F(x) =
0.5x
0.6
0.5x -0.6
0 < x < 1.2
1.2 < x < 2.4
2.4 < x < 3.2
0.1x -1
0.05x
10 < x < 15
15 < x < 20
Homework Problems
Stat 490C
146. * You are given that f(x) = (1/9)x2 for 0 < x < 3.
You are to simulate three observations from the distribution using the inversion
method. The follow three random numbers were generated from the uniform
distribution on [0,1]:
0.008
0.729
0.125
Using the three simulated observations, estimate the mean of the distribution.
147. * You are to simulate four observations from a binomial distribution with two
trials and probability of success of 0.30. The following random numbers are
generated from the uniform distribution on [0,1]:
0.91
0.21
0.72
0.48
Determine the number of simulated observations for which the number of
successes equals zero.
148. Kyle has an automobile insurance policy. The policy has a deductible of 500 for
each claim. Kyle is responsible for payment of the deductible.
The number of claims follows a Poison distribution with a mean of 2.
Automobile claims are distributed exponentially with a mean of 1000.
Kyle uses simulation to estimate the claims. A random number is first used to
calculate the number of claims. Then each claim is estimated using random
numbers using the inverse transformation method.
The random numbers generated from a uniform distribution on (0, 1) are 0.7, 0.1,
0.5, 0.8, 0.3, 0.7, 0.2.
Calculate the simulated amount that Kyle would have to pay in the first year.
Homework Problems
Stat 490C
149. * Insurance for a city’s snow removal costs covers four winter months.
You are given:
 There is a deductible of 10,000 per month.
 The insurer assumes that the city’s monthly costs are independent and
normally distributed with mean of 15,000 and standard deviation of 2000.
 To simulate four months of claim costs, the insurer uses the inversion
method (where small random numbers correspond to low costs).
 The four numbers drawn from the uniform distribution on [0,1] are:
0.5398
0.1151
0.0013
0.7881
Calculate the insurer’s simulated claim cost.
150. * Annual dental claims are modeled as a compound Poisson process where the
number of claims has mean of 2 and the loss amounts have a two-parameter
Pareto distribution with θ = 500 and α = 2.
An insurance pays 80% of the first 750 and 100% of annual losses in excess of
750.
You simulate the number of claims and loss amounts using the inversion method.
The random number to simulate the number of claims is 0.80. The random
numbers to simulate the amount of claims are 0.60, 0.25, 0.70, 0.10, and 0.80.
Calculate the simulated insurance claims for one year.
151. A sample of two selected from a uniform distribution over (1,U) produces the
following values:
3
7
You estimate U as the Max(X1, X2).
Estimate the Mean Square Error of your estimate of U using the bootstrap
method.
Homework Problems
Stat 490C
152. * Three observed values from the random variable X are:
1
1
4
You estimate the third central moment of X using the estimator:
g(X1, X2, X3) = 1/3 Σ(Xj - X )3
Determine the bootstrap estimate of the mean-squared error of g.
Homework Problems
Stat 490C
Answers
1.
2.
3.
4.
5.
Not Provided
¾
2
100/9999
a. 0.725
b. 0.093
6.
7.
8.
9.
a. 0.822
b. 0.244
No Answer Given
1/√5
a.
b.
c.
d.
(bk+1 – ak+1)/(b-a)(k+1)
(b+a)/2
(b-a)2/12
(b-d)/2
10.
a. θ/(α-1)
b. αθ2/[(α-1)2(α-2)]
c. [α/( α-2)]0.5
11.
a.
b.
c.
d.
θα
θ2 α
1/√α
2/√α
12.
a. θ
b. θ2
c. θ
13.
a.
b.
c.
d.
e.
f.
2.25
27/80
17/13
34/27
50/27
2.3811
Homework Problems
Stat 490C
Answers
14.
a.
b.
c.
d.
e.
f.
g.
7
5.8333
3.7333
3.1111
6.8888
7
5
15.
a. 4
b. 5.2
c. 2.28
d. 0.57
e. 0.8096
f. 2.145
g. 0.4z2 + 0.2z3 + 0.2z5 + 0.2z8
16. 5/3
17. 2500
18. 0.0244
19.
a. 1400
b. 26,973,333
c. 87.7%
20. 100
21. 12
22. i only. Be sure to indicate how to make the others true.
23. μ = 8.26 and σ = 2.00
24. 0.886227
25. 2 and 14/3
26. 7.2%
27. 0.6094
28. 2.6%
29.
a. 0.0498
b. 0.1494
c. 0.2240
d. 3
e. 3
30.
a. 0
b. 0.1572
c. 0.2358
d. 3.1572
e. 2.6609
Homework Problems
Stat 490C
Answers
31.
a.
b.
c.
d.
e.
0.5000
0.0786
0.1179
1.5786
3.8224
a.
b.
c.
d.
e.
0.2500
0.1875
0.1406
3
12
a.
b.
c.
d.
e.
0.0467
0.1866
0.3110
2.4
1.44
a.
b.
c.
d.
e.
0.0370
0.0741
0.0988
6
18
32.
33.
34.
35.
36.
37.
38.
39.
40.
a. 0
b. 0.5410
c. 0.2029
d. 2.1640
e. 3.9731
0.0165
7
Statement III is only true statement
0.2231
a. 4
b. 1.6943
c. 1.125
Homework Problems
Stat 490C
Answers
41.
a. fYL(y) = 1 – e-0.5, y = 0
fYL(y) = 0.001e-(0.001y + 0.5), y > 0
b. FYL(y) = 1 – e-0.5, y = 0
FYL(y) = 1 – e-(.001 y + 0.5), y > 0
c. SYL(y) = e-0.5, y = 0
SYL(y) = e-(.001 y + 0.5), y > 0
d. 1000e-0.5
e. 1 – e-0.5
42.
a. fYP(y) = 0.001e-0.001y
b. FYP(y) = 1 – e-.001 y
c. 1000
43.
a. fYL(y) = 1 – e-0.5, y = 0
fYL(y) = 0.001e-0.001y, y > 0
b. FYL(y) = 1 – e-0.5, y = 0
FYL(y) = 1 – e-.001 y, y > 0
c. SYL(y) = e-0.5, y = 0
SYL(y) = e-.001 y , y > 0
d. 1500e-0.5
e. 1 – 1.5e-0.5
44.
a. fYP(y) = 0.001e-0.001y+0.5
b. FYP(y) = 1 – e-.001 y+0.5
c. 1500
Homework Problems
Stat 490C
Answers
45.
a. 1100e-(0.5/1.1)
b. 1100
46. 2000e-400/θ
47. 0.165
48.
a. 4.0278
b. 4.39
c. 5.58
d. 4.48
49.
a. 38,500
b. 42,777.78
c. 641,083,333
d. 529,320,798
50.
a. 629.56
b. 939.19
51. 182.18
52. 1500
53.
a. (b-d)2/2b
b. [(b-d)3/3b] – [(b-d)2/2b]2
c. (b-d)/2
d. (b-d)2/12
54. 175
55. 71,833.62
56. 20
57. 1708.70
58. 705.06
59. 16,224
60. 333.33
61. 431.83
62. 2.5 and 23.75
63. 0.1587
64. 0.068
Homework Problems
Stat 490C
65.
66.
67.
68.
69.
70.
71.
x
fS(x)
0
1
2
3
4
5
6
0.024
0.130
0.280
0.280
0.180
0.086
0.020
0.151436
None provided
51.25
2.360894
22.60
a. 4,650
b. 11,377,500
c. 0.65(1-1000t)-3 + 0.2(1-1000t)-6 + 0.1(1-1000t)-9 + 0.05(1-1000t)-12
72. No Answer Provided
73. 0.2384
74.
a. Mean = 498.61
b. Mean = 499.85
75. 0.106978 and 0.060129
76. 0.1517
77. 76
78. 165
79. 0.15625
80. 0.406
81. 0.224
82. 92.16
83. 2.25
84. 5,727
85. 46.13
86. 2.1
87. 1975
88. 0.954
89. 0.30
90. 0.543
91. 0.2775
92. 0.6254
93. (n+8)/[18(n-1)2]
94. 151.52
95. 1/5
96. -0.24
Homework Problems
Stat 490C
97. 5
98. z = 2.0814; critical value = 2.33; Since 2.0814 is less than 2.33, we cannot reject
the null hypothesis; p = 0.0188
99.
x
p100(x)
x
F100(x)
Ĥ(x)
Ŝ(x)
x<1
0
0
1
1
0.06
1<x<2
0.06
0.060000
0.941765
2
0.02
2<x<4
0.08
0.081277
0.921939
4
0.03
4<x<6
0.11
0.113885
0.892360
6
0.04
6<x<7
0.15
0.158829
0.853142
7
0.04
7<x<8
0.19
0.205888
0.813924
8
0.05
8<x<9
0.24
0.267616
0.765201
9
0.07
9<x<10
0.31
0.359722
0.697871
10
0.06
10<x<11
0.37
0.446678
0.639750
11
0.07
11<x<12
0.44
0.557789
0.572473
12
0.07
12<x<13
0.51
0.682789
0.505206
13
0.48
13<x<22
0.99
1.662381
0.189687
22
0.01
x>22
1.00
2.662381
0.069782
Empirical Mean = 10.44 and Empirical Variance = 14.4064
100.
a. 0.04x for 0 < x < 10
0.15 + 0.025x for 10 < x < 20
0.25 + 0.02x for 20 < x < 30
Undefined for x > 30
b. 0.04 for 0 < x < 10
0.025 for 10 < x < 20
0.02 for 20 < x < 30
Undefined for x > 30
c. 8
101. 47.50 and 3958.333333333
102.
x
f(x)
x<0
0
0<x<1
3/60
1<x<2
7/60
2 < x < 2.8
9/60
2.8 < x < 3.0
17/60
3.0 < x < 3.5
13/60
3.5 < x < 4
23/60
4 < x < 4.8
18/60
4.8 < x < 5.5
10/60
5.5 < x
0
F(x)
0
3x/60
(7x-4)/60
(9x-8)/60
(17x-30.4)/60
(13x-18.4)/60
(23x-53.4)/60
(18x-33.4)/60
(10x+5)/60
1
Homework Problems
Stat 490C
103.
104.
105.
106.
107.
108.
109.
110.
111.
112.
113.
114.
115.
116.
117.
118.
119.
120.
121.
122.
123.
124.
125.
126.
127.
128.
129.
130.
131.
132.
133.
134.
135.
136.
137.
138.
139.
140.
141.
142.
143.
144.
0.53125
0.3
Mean = 24 and Variance = 151.50
Mean = 24 and Variance = 163.50
Mean = 24 and Variance = 378.00
α = 13.812 and θ = 272.89
α = 3.81889 and θ = 310.0782
10.8
X /( X  1)
1.8
18.4
1/6
234.91
378
2
3.002
1996.90
1.5
25
0.55
6
703
2.7386
7.5
0.6798
1.6436
m = 30 and q = 0.0573333333
0.28667
m = 26 and q = 0.066154
(1.50426, 1.93574)
0.1264
(1/3,0.2447)
All Statements are false.
0.2534, 0.61, Cannot reject,
10% => Reject at A2 > 1.933
5% => Reject at A2 > 2.492
1% => Reject at A2 > 3.857
B
0.5
0.1818
χ2 = 5.5100; critical value = 4.605; Reject H0
χ2 = 18.5; critical value = 7.779; Reject H0
6.6586
T = 2.197; critical value = 4.605
False; True; True
Homework Problems
Stat 490C
145.
146.
147.
148.
149.
150.
151.
152.
714.42, 2.4, 15
1.6
2
1105.36
14,400
630.79
4
4 8/9