Math 493 - Homework

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# Homework

## Homework 6, due Dec 9 at 2:07pm

Please do the following problems from the textbook:

DeGroot-Schervish 3.9, p187: 7, 14, 19
DeGroot-Schervish 4.1, p216: 7
DeGroot-Schervish 4.2, p224: 4, 5, 11
DeGroot-Schervish 4.3, p233: 5
DeGroot-Schervish 4.9, p272: 4

And the following additional problems:

A.  Let X and Y be independent continuous random variables with pdfs, and let phi and psi be strictly increasing, differentiable functions. Find the joint pdf of phi(X) and psi(Y). Conclude that phi(X) and psi(Y) are independent.

B.  If X is an exponential random variable with parameter c (recall this means pdf ce-cx for x > 0, zero elsewhere), then find the expected value and variance of X.

Hint for p187 #7: first find the pdf of -X2.

Hint for p224, #11: Example 4.2.9 is relevant to finding the expected value of Xi.

## Homework 5, due Nov 30 at 2:07pm

Please do the following problems from the textbook:

DeGroot-Schervish 3.4, p129: 10
DeGroot-Schervish 3.5, p141: 14, 15
DeGroot-Schervish 3.6, p151: 6
DeGroot-Schervish 3.7, p167: 4, 5
DeGroot-Schervish 3.8, p174: 2, 8, 11
DeGroot-Schervish 3.9, p187: 1, 6

And the following additional problems:

A.  You choose a point uniformly at random from the interior of the unit circle (the unit circle dartboard example from class). Let R be the radius and T the angle in a polar coordinates expression for this point. Find the joint cdf of R and T. Are R and T independent?

B.  If X is a random variable with cdf F, and phi is a strictly decreasing function on the support of F, then find the cdf of phi(X). If, in addition, X has a pdf and phi is differentiable, then find the pdf of phi(X).

C.  If X is a (continuous) uniform random variable on [-1,2], then find the cdf and pdf of X2.
Hint: It may be helpful to condition on the sign of X.

## Homework 4, due Nov 14 at 2:07pm

Please do the following problems from the textbook:

DeGroot-Schervish 3.1, p100: 9, 11
DeGroot-Schervish 3.2, p107: 6, 8, 10
DeGroot-Schervish 3.3, p116: 4, 6, 12, 17ab
DeGroot-Schervish 3.4, p129: 4, 8, 10