P/1 Exam problem

[edwinjaxfl]

Sharing a problem for practice. Here it goes:

Let X be a random variable with mean 3 and variance 2, and let Y be a random variable such that for every x, the conditional distribution of Y given X=x has a mean of x and a variance of x². What is the variance of the marginal distribution of Y? :goofy:

A) 5
B) 4
C) 13
D) 11
E) 2

[ecofreak11]

Sharing a problem for practice. Here it goes:

Let X be a random variable with mean 3 and variance 2, and let Y be a random variable such that for every x, the conditional distribution of Y given X=x has a mean of x and a variance of x². What is the variance of the marginal distribution of Y? :goofy:

A) 5
B) 4
C) 13
D) 11
E) 2

if you know that Yar(Y)=E(Var(Y|X))+Var(E(Y|X)), you can see that

Var(Y)=E(X^2)+Var(X)=11+2=13.

Otherwise, you should at least know that E(g(X))=E(E(g(X))), then
E(Y)=E(X)=3, E(Y^2)=E(E(Y^2|X))=E(2X^2)=22, and this gives Var(Y)=22-9=13.

[ChefJive]

if you know that Yar(Y)=E(Var(Y|X))+Var(E(Y|X)), you can see that

Var(Y)=E(X^2)+Var(X)=11+2=13.

Otherwise, you should at least know that E(g(X))=E(E(g(X))), then
E(Y)=E(X)=3, E(Y^2)=E(E(Y^2|X))=E(2X^2)=22, and this gives Var(Y)=22-9=13.

Yarr, variance for pirates.

[Apollon]

That is a really cool problem. I had no idea how to work it.

[edwinjaxfl]

That's the right answer. The Actex book states that we might not see this type of problem on the P/1 test but that we might on the (MLC) modeling test or the (C) construction test. I thought it was good to work out anyway.