Probability

STAT 4351: Homework 8 Due by July 27, 2020

Yuly Koshevnik

First Name Last Name

Problem 1 2 3 4 5 6 7 8 Total

Points

Maximum 10 10 10 10 15 15 15 15 100

Please submit the front page with your solutions!

1

Problem 1 [10 points = 5 + 5]

A continuous non-negative random variable, Y has density function

fY (y) = 4y2 · exp(−2y)

Given Y = y, another non-negative random variable, W has conditional density

fW |Y (w|y) = y2 ·w · exp(−wy)

1. Determine the marginal density for W, fW (w) =

2. Derive the conditional density for (Y |W = w), fY |W (y|w) =

Solution

2

Problem 2 [10 points = 4 + 6]

A continuous non-negative random variable, Y has density function

fY (y) = 4y2 · exp(−2y)

Given Y = y, another non-negative random variable, W has conditional density

fW |Y (w|y) = y2 ·w · exp(−wy)

Evaluate marginal first and second moments listed below

1. E [W ] =

2. Var [W ] =

Solution

3

Problem 3 [10 points = 5 + 5]

A continuous non-negative random variable, Y has density function

fY (y) = 4y2 · exp(−2y)

Given Y = y, another non-negative random variable, W has conditional density

fW |Y (w|y) = y2 ·w · exp(−wy)

Evaluate conditional moments listed below.

1. E [Y |W = w] =

2. Var [Y |W = w] =

Solution

4

Problem 4: [10 points = 5 + 5]

Two non-negative random variables, T and W, are independent with density functions

fT (t) = 37

720 t6 · exp(−3t) and fW w =

35

24 w4 · exp(−3w)

Derive expected values listed below.

1.

E [ W

T

] =

2.

E [ T

W

] =

Solution

5

Problem 5 [15 points = 5 + 5 + 5]

A sample X = {Xi : 1 ≤ i ≤ 7} of size n = 7 was drawn from a uniform distribution with density

f(x) = 1

5 for (0 < x < 5)

Consider the third order statistic, W = X[3] based on this sample.

1. Derive density function, fW (w) =

2. Evaluate expectation, E [W ] =

3. Find the variance, Var [W ]

Please, simplify your answers!

Solution

6

Problem 6 [15 points = 5 + 5 + 5]

A sample X = {Xi : 1 ≤ i ≤ 7} of size n = 7 was drawn from a uniform distribution with density

f(x) = 1

5 for (0 < x < 5)

Consider the third and seventh order statistics, W = X[3] and Y = X[7]

1. Derive the density function for the transformed variable,

T = Y −W

W

2. Determine E [T ]

3. Find expectation of the ratio,

Z = Y

W

Solution

7

Problem 7 [15 points = 5 + 5 + 5]

A three-dimensional random vector X = (X1, X2, X3) has independent components such that

X1 ∼ N [1, 2], X2 ∼ N [−2, 4], and X3 ∼ N [1, 3]

Consider a bivariate random vector Y = (Y1, Y2) with components defined as follows:

Y1 = X1 + X2 + X3 and Y2 = X1 −X3

1. Derive expectations, E [Y1] and E [Y2]

2. Evaluate variances, Var [Y1] and Var [Y2]

3. Determine the covariance, Cov [Y1, Y2]

Solution

8

Problem 8 [15 points = 5 + 5 + 5]

A non-negative continuous random variable, Y has density function defined as follows:

fY (y) = 39

8! ·y8 · exp(−3y)

Given Y = y, a non-negative integer-valued random variable, N has probability mass function

fN|Y (n|y) = P [N = n|Y = y] = yn

n! · exp(−y)

1. Evaluate marginal moments E [N] = and Var [N] =

2. Derive the conditional density function for (Y |N)

3. Evaluate the moments for (Y |N = 3) E [Y |N = 3] =

and Var [Y |N = 3]

Solution

9