W10 - HW Solutions

01

(1) Compute :

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(2) Compute :

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(3) Compute :

02

For :

So:

03

(a)

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(b)

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(c)

04

(a)

Conditional distribution:

Compute :

Therefore:

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(b)

Therefore:

05

(a)

Obtain joint from conditional and marginal:

Therefore:

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(b)

Conditioning the other way:

Marginal distribution of :

Therefore:

06

Find possible combinations of and .

• For a given value of , there are possible options for .
• Thus, in total, there are possible combinations.
• Since , we have that .

(a)

We have that

Thus,

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(b)

(1) State formula

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(2) State final answer

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(c)

Compute expectation:

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(d)

Find formula for expectation.

07

Integrate joint PDF and solve for .

(a)

Integrate joint PDF with respect to to obtain

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(b)

Use formula for conditional distribution.

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(c)

(1) Plug in into the formula found in part (b)

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(2) Integrate to find expectation.

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(d)

Integrate formula in part (b) with respect to .

08

(1) State formula for expectation given .

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(2) Note that is fixed inside the conditional expectation.

• Treat as inside integral.

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(3) Prove the second formula.

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(4) State final answer.

09

Let and .

10

(1) Find .

Note that this follows a binomial distribution with parameters .

Thus,

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(2) Find using iterated expectation.

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(3) Find

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(4) Find covariance

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(5) State final conclusions. Since the covariance is positive, and are positively correlated.