Question

Suppose X and Y are {0, 1}-valued random variables with joint probability mass function given in the table below p(x, y) y 0 1 x 0 0.3 0.2 1 0.1 0.4 a. Determine E(Y |X = x) and var(Y |X = x). b. Use part a) to find E(Y ), and compare with the result using the marginal distribution of Y . c. Use part a) to find var(Y ), and compare with the result using the marginal distribution of Y .

Answer 1

a)

When X=0:

Following table shows the calculations for mean and variance

So

When X=1:

Following table shows the calculations for mean and variance

So

b)

Following table shows the calculations for Y:

So,

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Following table shows the calculations E[E(Y|X=x)]:

So,

c)

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Therefore