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W11 - Homework

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Conditional distribution

01

Conditional density from joint density

Suppose that and have joint probability density given by:

(a) Compute , for .

(b) Compute .

02

From conditional to joint, and back again

Suppose we have the following data about random variables and :

(a) Find the joint distribution .

(b) Find .

Conditional expectation

03

Conditional distribution and expectation from joint PMF

Suppose that and have the following joint PMF:

Notice that the possibilities for depend on the choice of .

First, show that must be true. Then compute:

(a) (b) (c) (d)

04

Conditional distribution and expectation from joint PDF

Suppose that and have the following joint PDF:

Notice that the range of possibilities for depends on the choice of .

First, show that must be true. Then compute:

(a) (b) (c) (d)

05

“Plug In” Expectation Identity

Suppose is a function, and and are two random variables.

Verify this formula in the continuous case, using the definitions:

Using that formula, prove this formula:
for two functions and and random variables and . Notice that here the expectations are viewed as random variables.

Hint for second question: Both sides are functions of . Write these functions as and and check equality of the functions.

06

Expectation Multiplication Rule

Prove the following identity using Iterated Expectation along with the previous exercise:

Note: The solution is short once you find it – please clearly identify your choices for and functions.

07

How many customers buy a cake?

Let count the number of customers that visit a bakery on a random day, and assume .

Let count the number of customers that make a purchase. Each customer entering the bakery smells the cakes, and this produces a probability of buying a cake for that customer. The customers are independent.

Find . Are and positively or negatively correlated?

Hint: Compute , and use this to compute in terms of . Now deduce using Iterated Expectation. Finally, compute using the Expectation Multiplication Rule from the previous exercise. Now put everything together to find .