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Exam 2
1.
Fall02
A sample of 150 items is inspected from a population of 1000. It is believed that 23% of the
population is overweight. Which of the following is the expected number of overweights to be found
in the sample?
(a) 3.45
(b) 23
(c) 34.5
(d) 230
(e) 3450
We are interested in paint drying times. The paint drying times are normally distributed with the mean 120
minutes and standard deviation 15 minutes. Answer the following 6 questions using this information.
2.
Which of the following is the 33th percentile of the paint drying time?
(a) 105 minutes
(b) 113.4 minutes
(c) 115.05 minutes
(d) 120 minutes
(e) 126.6 minutes
3.
Which of the following is the probability that the paint drying time is more than 2.5 hours?
(a) 0
(b) 0.0228
(c) 0.4772
(d) 0.9772
(e) 1
4.
Which of the following is the median paint drying time?
(a) 0 minutes
(b) 15 minutes
(c) 60 minutes
(d) 120 minutes
(e) 135 minutes
5.
Let X1,…,X40 be the random sample of paint drying times. Which of the following is expected total
drying times?
(a) 0 minutes
(b) 24 minutes
(c) 120 minutes
(d) 2400 minutes
(e) 4800 minutes
6.
Let X1,…,X40 be the random sample of paint drying times. Which of the following is the standard
deviation of total drying times?
(a) 15 minutes
(b) 95 minutes
(c) 600 minutes
(d) 1050 minutes
(e) 9000 minutes
7.
Let X1,…,X40 be the random sample of paint drying times. Which of the following is probability of
average paint drying time being less than 123 minutes?
(a) 0.1020
(b) 0.2351
(c) 0.4207
(d) 0.5793
(e) 0.8980
Exam 2
8.
Fall02
Cars arrive at a toll plaza according to a Poisson process. Which of the following is the probability of
exactly 5 arrivals in 1 minute when the average rate of arrivals per minute is 2?
(a) 0.0125
(b) 0.0361
(c) 0.0673
(d) 0.1098
(e) 0.1674
The time between accidents at a congested intersection has an exponential distribution with =2/hour.
Answer the following 3 questions using this information.
9.
Which of the following is the expected time between accidents?
(a) 0.5 hour
(b) 1 hour
(c) 1.5 hours
(d) 2 hours
(e) 2.5 hours
10. Which of the following is the standard deviation in time between accidents?
(a) 0.5 hour
(b) 1 hour
(c) 1.5 hours
(d) 2 hours
(e) 2.5 hours
11. Which of the following is the 25th percentile in time between accidents?
(a) 0.1439 hour
(b) 0.2500 hour
(c) 0.3068 hour
(d) 0.6932 hour
(e) 0.7500 hour
12. How would you determine if the given data come from a Poisson distribution?
(a) I would graph data, x versus any p(x) to see if they make a symmetric graph.
(b) I would graph data, x versus any p(x) to see if they make a 45 line.
(c) I would graph the ordered data with their expected Poisson values to see if they make a 45 line.
(d) I would graph the ordered data with their expected normality values to see if they make a 45 line.
(e) I would graph data, x versus Poisson p(x) to see if they make a symmetric graph.
Use the following choices to answer the next 3 questions.
(a) Binomial Distribution
(b) Hypergeometric Distribution
(c) Poisson distribution
(d) Exponential Distribution
(e) Normal distribution
13. A quality control inspector rejects any shipment of printed circuit boards whenever 3 or more
defectives are found in a sample of 20 boards tested. Which distribution would you use to compute the
probability of rejecting the shipment when the proportion of defectives is 0.05?
14. The n power cells in a satellite will be arranged in parallel and will fail at a rate of 0.01 per day. They
have independent lifetimes. Which of the following would you use to compute the probability that any
specific cell will fail on or before 100 days?
Exam 2
Fall02
15. The freshman class at an engineering school contains 300 students. 10% of them have reserved space
in the civil engineering option. Which distribution would you use to compute the probability that there
will be 10% or less civil engineering majors within the first 20 names from an alphabetical roster?
A joint pdf for x and y is defined by
 y  x,
f ( x, y )  
0,
if
0  x 1
otherwise
and
0 y2
Use this information to answer the following 4 questions.
16. Which of the following is the marginal pdf for X?
(a) 0.5-x when 0<x<1
(b) y-0.5 when 0<y<2
(c) 2(1-x) when 0<x<1
(d) 2(y-1) when 0<y<2
(e) y-1 when 0<y<2
17. If the marginal pdf for Y is f(y)=y-0.5, 0<y<2, are X and Y independent?
(a) No. f(x,y)=f(x)f(y) for all x and y
(b) Yes. f(x,y)=f(x)f(y) for all x and y
(c) No. f(x,y)f(x)f(y) for all x and y
(d) Yes. f(x,y) f(x)f(y) for all x and y
(e) No. f(x|y)=f(x) for all x and y
18. If the marginal pdf for Y is f(y)=y-0.5, 0<y<2, which of the following is the expected value of Y?
(a) 5/3
(b) 4/3
(c) 1
(d) 2/3
(e) 1/3
19. If the E(X)=1/3 and E(XY)=4/6 then which of the following is true?
(a) X and Y are positively related
(b) X and Y are negatively related
(c) X and Y are unrelated
(d) X and Y are independent
(e) There is no way to answer this question
20. If E(X)=3 and Var(X)=4, which of the following is the variance of Y=(X-1)/2?
(a) 0.5
(b) 1
(c) 1.5
(d) 1.75
(e) 2
21. Let Z be a standard normal random variable. Compute P(-0.45  Z 0)
(a) 0.1043
(b) 0.1736
(c) 0.3264
(d) 0.6736
(e) 0.8264
Exam 2
Fall02
Let X be the number of packages being mailed by a randomly selected customer at a certain shipping
facility. The distribution of X is
X
1
2
3
4
p(x)
0.4
0.3
0.2
0.1
Consider a random sample of size 2 (2 customers). The joint probability mass function for those two
customers (x1 and x2) are
x2 1
2
3
4
Total
x1
1
0.16 0.12 0.08 0.04 0.40
2
0.12 0.09 0.06 0.03 0.30
3
0.08 0.06 0.04 0.02 0.20
4
0.04 0.03 0.02 0.01 0.10
Total
0.40 0.30 0.30 0.10 1
Answer the following 4 questions using the information above.
22. Which of the following is the P(min(X1,X2)=3)?
(a) 0
(b) 0.02
(c) 0.04
(d) 0.06
(e) 0.08
23. Which of the following is the P(X1-X2=0)?
(a) 0.01
(b) 0.04
(c) 0.09
(d) 0.16
(e) 0.30
24. Which of the following is the expected number of packages being mailed by a randomly selected
customer, E(X)?
(a) 1.5
(b) 2
(c) 2.5
(d) 3
(e) 3.5
25. Which of the following is the P(X12,X22)?
(a) 0.16
(b) 0.24
(c) 0.39
(d) 0.49
(e) 0.54
Answer Key:
1.C
2.B
13.A 14.D
25.D
3.B
15.B
4.D
16.C
5.E
17.C
6.B
18.A
7.E
19.A
8.B
20.B
9.A
21.B
Formulas
Discrete probability distribution, p(x) is legitimate if
 0  p(x)=P(X=x)  1 for all x where X is a discrete random variable (r.v).

 p ( x)  1
all x
10.A
22.E
11.A
23.E
12.C
24.B
Exam 2
Fall02
Cumulative Distribution Function, (CDF) for discrete X:
F ( x)  P( X  x) 
x
 P( X  y )
for all y
P(a  X  b)  F (b)  F (a  1) and P( X  a)  F (a)  F (a  1) where a and b are
The expected value of a discrete variable, X :
 x  p ( x)
  E( X ) 
for all x
The expected value of a function of discrete variables, h(x) :
The variance of a discrete variable, X: 2 = Var(X) =
E( X 2 ) 
x
2
E (h( x)) 
 h( x )  p ( x )
for all x
2
E[( X   ) ]  E ( X 2 )   2
where
 p ( x)
for all x
The standard deviation of numerical variable, X:  =
2
Bernoulli Distribution: It is based on Bernoulli trial (an experiment with two, and only two, possible
outcomes). A r.v. X has a Bernoulli(p) distribution if
P(X=x)=
p x  (1  p)1x ,
x  0,1 and 0p1. where E(X) = p and Var(X) = p(1-p)
Binomial Distribution: Approximate probability model for sampling without replacement from a finite
dichotomous population. X~Binomial(n,p).
 n fixed trials
 each trial is identical and results in success or failure
 independent trials
 the probability of success (p) is constant from trial to trial
 X is the number of successes among n trials
n
P( X  x)     p x  (1  p) nx ,
 x
E(X) = np
and
x  0,1,2,...., n
Var(X) = np(1-p)
Hypergeometric Distribution: Exact probability model for the number of successes in the sample.
X~Hyper(M,N,n)
 M  N  M 
 

x
n

x
,
P( X  x)   
N
 
n 
max( 0, n  N  M )  x  min( n, M )
Let X be the number of successes in the sample, n be the sample size, N be the population size, and M be
the number of successes in the population
M
M
where
is the proportion of successes in the population.
N
N
N n
N n
M  M
 n   1   where
Var(X) =
is the finite population correction factor.
N 1
N 1
N 
N
E(X) = n 
Poisson Distribution: The probability of an arrival is proportional to the length of waiting time.
P( X  x) 
e   x
, x  0,1,2,3,...........,
x!
 0
: intensity parameter (mean rate, expected number of occurrences).
Exam 2
Fall02
X : number of occurrences per given period

i
i 0
i!
e 

Note that
and E(X)=Var(X)=.
Continuous probability distribution, p(x) is legitimate
(i)
f(x)  0, for all x

 f ( x)dx  1 = area under the entire graph of f(x).
(ii)

x
Cumulative Distribution Function for continuous X (cdf) : F(x)= P ( X  x) 
 f ( y)dy.

F(-)=0, F()=1, P(a  X  b)  P(a  X  b)  F (b)  F (a ) , P(X>a)=P(Xa)=1-F(a)
Obtaining f(x) from F(x) : If X is a continuous r.v. with pdf f(x) and cdf F(x), then at every x at which the
derivative exists, F`(x)=f(x).
Percentile of a continuous distribution: Let p be a number between 0 and 1. the (100p)th percentile of
the
distribution
of
a
continuous
r.v.
X,
denoted
by
r(p),
is
defined
by
r ( p)
 f ( y)dy.
p  F (r ( p ))  P( X  r ( p)) 
where P(Xmedian)=P(X>median)=0.50

Expected value for the continuous random variable, X:
  E( X ) 

 x  f ( x)dx.


Expected value for the function of continuous random variables, h(x): E ( h( x)) 
 h( x)  f ( x)dx.

Variance for the continuous random variable, X:
  Var( X )  E ( X )  
2
2

2
where
E( X ) 
2
x
2
 f ( x)dx.

1
,
ba
X 
Uniform Distribution: X ~U[a,b] then f ( x) 
Normal Distribution: X ~ N ( , 2 ) and z 

axb
~ N (0 , 1)
Normal Approximation to the Binomial Distribution:
Let X be a binomial r.v. based on n trials with success probability p. Then if the binomial probability
histogram is not too skewed, X has approximately a normal distribution with  = np and
 x  0.5  np 
 (check if np10 and n(1-p)10 to use the

 np(1  p) 
  np(1  p) then P( X  x)  
formula).
The Gamma Distribution:X~ Gamma (  ,  ) then (1 / 2) 

( )   x  1 e  x dx,
 0
0
 (  1)  (  1),
 1
 (  1)!,
 is any positive integer
 , E ( X )   , Var ( X )   2
Exam 2
Fall02


If =1 then it is called standard gamma distribution.
When the random variable is a standard gamma r.v. then the cdf is called the incomplete gamma
function (Appendix Table A.4). F(x;,)=F(x/;)

If  = 1 then it is Exponential(=1/).
f ( x)    e    x , x>0 and P( X  x*)  1  e x* .
Discrete Data: If p(x,y) is the joint pmf for x and y, the marginal pmf for x can be computed as
 p( x, y) and the marginal pmf for y can be computed as  p( x, y) .
y
E(X)=
 x  p( x, y)   x  p( x) and E ( X
x
E(XY)=
y
x
)   x 2  p( x, y )   x 2  p( x)
x
 x  y  p( x, y)
x
x
2
and
E(h(X,Y))=
y
x
 h( x, y)  p( x, y)
x
p(x | y)=p(x,y) / p(y) where p(y)>0 and E(X|Y)=
y
y
 x  p( x | y )
x
Continuous Data: If f(x,y) is the joint pdf for x and y, the marginal pdf for x can be computed as
 p( x, y )dy and the marginal pdf for y can be computed as  p( x, y)dx .
y
E(X)=
x
  x  f ( x, y)dydx   x  f ( x)dx
x y
E(XY)=
and E ( X ) 
2
x
  x  y  f ( x, y)dydx
x
2
 f ( x, y )dydx   x 2  f ( x, y )dx
x y
and
E(h(X,Y))=
x y
x
  h( x, y)  f ( x, y)dydx
x y
f(x | y)=f(x,y) / f(y) where f(y)>0 and E(X|Y)=
 x  f ( x | y)dx
x
For both discrete and continuous,

Var(X)= E ( X )  E ( X )
2
2
, Cov(X,Y)=E(XY) -E(X)E(Y) and
Cov( X , Y )
Corr(X,Y)=
Var ( X ) Var (Y ))
n
n
 n
 n
Var  ai X i    ai2Var ( X i )  2 ai a j Cov( X i , X j ) , i<j.
i 1 j 1
 i 1
 i 1
Random Sample:The random variables X1, X2, ….,Xn are said to form a random sample of size n if
(i) The Xi's are independent random variables.
(ii) Every Xi's has the same probability distribution.
If X1, X2, ….,Xn are said to form a random sample of size n with the mean  and the variance 2, the sampling
n
_
distribution of
x
has the mean  and the variance 2/n, the sampling distribution of
x
i 1
variance n2, and so on.
i
has the mean n and the