MTH 451 Summer 2013
Problem Set 11
1. If the joint pdf of two random variables X and Y is given by
f (x, y ) =
1
(2x
4
0
+ y ) for 0 < x < 1, 0 < y < 2
elsewhere
(a) Find cov(X, Y )
(b) Find the conditional mean of X given Y = 1 . (You already found the m
MTH 451 Summer 2013
Problem Set 9
1. If the pdf of a random variable X is given by
x
2
1
f (x) =
2
3x
2
0
for 0 < x < 1
1<x2
2<x<3
elsewhere
Find the expected value of g (X ) = X 2 5X + 3
2. If the joint pdf of two random variables X and Y is
f (x, y ) =
MTH 451 Summer 2013
Problem Set 10
1. If the pdf of a random variable X is given by
f ( x) =
Find , 2 and variance of X .
x
2
for 0 < x < 2
0 elsewhere
MTH 451 Summer 2013
Problem Set 12
1. Find the MGF for the uniform discrete distribution. (Note: You will need the formula
for sum of a geometric series with nite terms to complete this).
2. A biased coin is tossed 6 times. The probability of heads on any
MTH 451 Summer 2013
Problem Set 12
1. The MGF of the geometric distribution is given by (without proof)
MX (t) =
et
.
1 et (1 )
Use dierentiation to show that (convincingly) the mean =
1
. (Please show all steps clearly)
2
1
and the variance 2 =
2. If the
MTH 451 Summer 2013
Problem Set 1
Name:
1. In a group of 10 people each person has a dierent birthday. How many dierent arrangements of these birthdays are there (assuming each year has 365 days)?
2. We randomly select 3 balls from an urn with 365 balls,
MTH 451 Summer 2013
Problem Set 6
1. The pdf of a random variable X is given by
f (x) =
k
x
0
for 0 < x < 4
elsewhere
(a) Find k
(b) Find P (X < 1 ) and P (X > 1)
4
(c) Find the corresponding distribution function
2. The distribution function of a random
MTH 451 Summer 2013
Problem Set 5
1. Find the distribution function of the random variable that has the probability distribution for x = 1, 2, 3, 4, 5
x
f (x) =
15
2. Find the value of c so that the function f (x) = cx2 can be a probability distribution f
MTH 451 Summer 2013
Problem Set 4
1. In a recent political survey of 200 people in Texas, 120 people approve Presidents second term, 40 said they dont approve and 40 said they dont care. Among these the
probabilities are 0.15, 0.89 and 0.6 respectively th
MTH 451 Summer 2013
Problem Set 2
Note: You do not have to nd numerical values of permutations and combinations.
1. In class we calculated the probability of getting a full house in a poker game. Another
possibility is four of a kind which means out of th
MTH 451 Summer 2013
Problem Set 3
1. A bicycle company manufactures its product in two plants X and Y. Plant X manufactures 60% of the bicycles. 4% of the bikes from X and 5% of the bikes from Y are
defective.
(a) Find the probability that a bike is defec
MTH 451 Summer 2013
Problem Set 7
1. Find the value of k such that the joint pdf of two random variables X and Y is given by
f (x, y ) =
k x(x y ) for 0 < x < 1, x < y < x
0
elsewhere
2. If the joint pdf of two random variables X and Y is
f (x, y ) =
2 fo
MTH 451 Summer 2013
Problem Set 8
1. If the joint pdf of two random variables X and Y is given by
f (x, y ) =
1
(2x
4
+ y ) for 0 < x < 1, 0 < y < 2
elsewhere
0
(a) Find the marginal density of X
(b) Find the conditional density (not asking for probabilit