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PSTAT W120A, Summer 2016 Assignment 6 Due Wednesday Sep. 7 Directions: Please write up your solutions and submit them on GauchoSpace by the end of the day

on Wednesday. We prefer the submission to be single pdf file if possible. Bonus points will be awarded

for organization and neatness. Lesson 21 Conditional Expectations

1. Suppose that R and S have a joint probability mass function

S R 0

1

2

3

4 -1 0 1

10

1

20 1

10

1

10

1

20 0

0

0 0

0 1

0

1

20

1

10

1

10 0 2

0

0

1

20

1

10

1

5 (a) Calculate the conditional expectation of S | R = 2.

(b) Calculate the conditional expectation R2 given that S = 0.

(c) Calculate the expectation of RS given that R = 3. Lesson 22 Continuous Conditional Distributions

2. The joint density for X and Y is

2x + 1

2

for 0 ? x ? 1 and 0 ? y ? 1. Find the conditional density of X given that Y = y.

f (x, y) = 3. The joint distribution for X and Y has a sample space on the triangle 0 ? x ? 2, 0 ? y ?

density

f (x, y) = 3y x

2 with (a) Calculate the marginal density of X at X = 1.

(b) Calculate the conditional density of Y given that X = 1.

4. Suppose that we observe the following mixed model.

? First, X is generated by a Poisson distribution with ? = 5.

? Then, given that X = k, Y has a Gamma distribution with rate 10 and shape parameter k.

The conditional density for Y is thus

f (y | x) = 10x x?1 ?10y

y

e

?(x) for y ? 0. Calculate the expected value of Y .

Hint: You can use that the expected value of a Gamma distribution with shape parameter r and

rate ? has expected value E(Y ) = ?r . 1 of 2 PSTAT W120A, Summer 2016 Assignment 6 Due Wednesday Sep. 7 Lesson 23 Covariance

5. Calculate the Cov(R, S) from problem 1.

6. Suppose that we have a box with 3 white marbles, 3 green marbles, and 3 red marbles in it. We

choose three marbles without replacement from the box and let W = the number of white marbles

chosen and G =the number of green marbles chosen.

(a) Calculate the Var(W ).

(b) Calculate the Cov(W, G). 2 of 2

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