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Applied Probability with R
Home Work 1
1. If P (A ∪ B) = 0.7 and P (A ∪ B c ) = 0.9, then P (A) =?
2. Among 25 BSc Math (Hons.) students in a college 15 are good in Algebra, 16 are good in Calculus and 12 are
good in both. If one of the student is chosen at random, then what is the probability that she or he is good in
neither?
3. A professor have 3 research students viz., A, B, C. She assigns 40% of the problem to A of which A solves 3%
wrongly. She assigns 25% of the problem to B of which B solves 5% wrongly. She assigns 35% of the problems
to C of which C solves 4% wrongly. What is the probability that from the group of wrongly solved problems;
if a problem is selected randomly then it was actually assigned to A?
4. An urn has 6 red and 4 black balls if 5 balls are taken out of the urn at random with replacement then what
is the probability that there will be 3 red and 2 black balls?
5. An urn has 6 red and 4 black balls if 5 balls are taken out of the urn at random without replacement then what
is the probability that there will be 3 red and 2 black balls?
6. The probability
a family will have k children is αpk , k = 1, 2, ... the probability that it will have no child
P∞ that
k
being 1 − α k=1 p . Assuming that the sex-ratio at birth is 1:1. Obtain the probability that the family will
have x sons.
n
7. Suppose the probability that certain species lays n eggs is exp{−λ} λn! , n = 0, 1, 2, .... Also p is the probability
for an egg to develop. We assume that different eggs develop or donot develop is indepent of each other. Find
the probability that out of n eggs laid exactly k will develop.
8. Probability of an individual coal miner being killed during a year is
200 miners, there will be at least one fatal accident in a year.
1
2400 .
Calculate that in a mine employing
9. Suppose an insurer charges Rs 1000 as premium for a personal accident policy for sum-assured of Rs 100000.
That is if a claim takes place then the company will pay Rs. 100000 to the claimant. Suppose in the city
typical rate of accident is 1 out of 1000. If company sell an insurance then what is the expected revenue of the
company from that sell?
1
10. (Continuation of previous problem): If in a year the company sells 500 insurance and earn Rs 1000 from
each, then company’s revenue will be Rs 500000. Clearly if 5 or more than 5 claim takes place then that year
the insurance company will make a loss. What is the probability that the company will make a loss.
11. (Continuation of previous problem): However, the rate of accident in the city increases to 5 out of 1000.
Then recalculate the probability of that the company will make a loss. Clearly because of the increased rate of
accident now company faces higher probability of making loss. As an analyst what will be your advice so that
comany could keep the probability of making loss below 0.01! Be specific in your suggetion - so that probability
of comany’s loss is less than 0.01.
12. Donated blood is screened for HIV positive. Suppose the test has 99% accuracy, and the test that one in tes
thousand people in the same age group are HIV positive. The test has a 5% false-positive rating, as well.
Suppose the test screens someone as positive. What is the probability that the patient really is HIV positive?
13. Suppose a fair coin is tossed twice. Following 3 events are considered. A = {HH, HT }, B = {HH, T H} and
C = {HH, T T }. Show that 3 events are pairwise independent. But not completely independent.
14. If X ∼ N (µ, σ 2 ) then find its moment generating functions.
15. If X ∼ P oisson(λ) then find its moment generating functions.
16. Show that
∞
X
x2 exp{−λ}
x=0
λx
= λ + λ2
x!
17. Suppose X ∼ Bin(n, p), Y ∼ Bin(m, p) and X and Y are independent then show X + Y ∼ Bin(n + m, p)
iid
18. Suppose Xi ∼ Bin(ni , p), ∀i = 1, ..., n show that Sn =
iid
19. Suppose Xi ∼ Bin(1, p), ∀i = 1, ..., n show that Sn =
Pn
i=1
Pn
iid
iid
21. Suppose Xi ∼ P oisson(λ), ∀i = 1, ..., n show that Sn =
Xi ∼ Bin(n, p)
i=1
20. Suppose Xi ∼ P oisson(λi ), ∀i = 1, ..., n show that Sn =
Pn
Xi ∼ Bin( i=1 ni , p)
Pn
i=1
Pn
i=1
Pn
Xi ∼ P oisson( i=1 λi )
Xi ∼ P oisson(nλ)
Gamma Distribution A random variable X is defined to have a Gamma distribution with parameter (α, β)
(α > 0, β > 0) if its pdf is given as
f (x|α, β) =
αβ −αx β−1
e
x
,
Γ(β)
where
Z
Γ(β) =
0 < x < ∞,
∞
e−t tβ−1 dt (β > 0)
0
2
22. Show that if X ∼ Gamma(α, β) then the mgf of X is
MX (t) = E(e
23. If Xj ∼ Gamma(α, βj ), j = 1, 2..., n, then Sn =
Pn
tX
j=1
)=
α
α−t
β
Xj ∼ Gamma(α,
Pn
j=1
βj ).
24. From 6 positive and 8 negative numbers, 4 numbers are chosen at random (without replacement) and multiplied.
What is the probability that the product is positive?
25. From 6 positive and 8 negative numbers, 4 numbers are chosen at random (without replavement). What is the
probability there will be two positive and two negative numbers?
26. A box contains 4 bad and 6 good tubes. Two are drwan out from the box one at a time. One of them is tested
and found good. What is the probability that the other one is also good?
27. A product is manufactured by 3 machines viz., A, B, C. A produces half of the total production. B and C
produces equal number. 2% of the products produced by A and B are defective and 4% of products produced
by C are defective. What is the probability that a randomly selected product is defective?
28. Probability of an individual will default on his/her credit is
there will be at least one credit default in a year.
1
100 .
Calculate that out of 200 debtors of the bank,
29. What is the expected number of credit default that manager can expect over the year?
30. What is the standard deviation of the credit default?
31. Suppose in the loan program the bank earn 10% of every Rs 100000 that are lent to the debtor. If the person
default on his/her credit then Bank loss the whole money. What is the expected revenue of the Bank from
every Rs 100000.
32. There are 3 fair coins and 1 Sholay coin with ‘head’ on both sides. A coin is chosen at random and tossed 4
times. If ‘head’ occurs all the 4 times, what is the probability that the false coin has been chosen and used?
33. A basket has 10 red and 5 green apples. If 5 apples are taken out of the basket at random with replacement
then what is the probability that there will be 3 red and 2 green apples?
34. A basket has 10 red and 5 green apples. If 5 apples are taken out of the basket at random without replacement
then what is the probability that there will be 3 red and 2 green apples?
iid
35. Suppose Xi ∼ N (µ, σ 2 ), ∀i = 1, 2, ..., n.
Pn
(a) Show that Sn = i=1 Xi ∼ N (nµ, nσ 2 )
(b) Show that X̄ =
Sn
n
2
∼ N (µ, σn )
3
(c) Show that limn→∞ P (|X̄ − µ| <= ) = 1 for any known > 0.
iid
36. Suppose Xi ∼ Binomial(1, p),
∀i = 1, 2, ..., n.
(a) Find the moment generating functions of X1 .
Pn
(b) Show that Sn = i=1 Xi ∼ Binomial(n, p)
(c) Show that
lim P
n→∞
Sn
− p| ≤ |
n
=1
for any known > 0.
37. Consider the function f (x) = exp{−λx} , where 0 < x < ∞ and λ > 0
(a) Show that the function is not probability density function (pdf).
(b) Make the necessary changes,so that the function becomes a pdf.
(c) Find the mean and variance of new pdf.
38. A continuous random variable has pdf f (x) = 3x2 , where 0 < x < 1 Find a and b such that
(a) P (X ≤ a) = P (X > a)
(b) P (X > b) = 0.5
39. Suppose X: the amount of profit a company is expecting in million dollars
X
P (X)
-2
0.1
-1
k
0
0.2
(a) Find k.
(b) Evaluate P (X < 2) and P (−2 < x < 2)
(c) Evaluate mean of X.
4
1
2k
2
0.3
3
3k