(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose the distribution of X2 conditional on X1=x1 is N(x1,x1^{2}), and that the marginal distribution of X1 is U(0,1). Find the mean and variance of X2.

2. Relevant equations

Theorem: [tex]E(X_{2})=E_{1}(E_{2|1}(X_{2}|X_{1}))[/tex]

[tex]Var(X_{2})=V_{1}(E_{2|1}(X_{2}|X_{1}))+E_{1}(V_{2|1}(X_{2}|X_{1}))[/tex]

3. The attempt at a solution

If I understand the above right, which I'm not sure I do, in words, the first part says that the expected value of X2 is going to be the expected value of the conditional function. Here, the expected value of X2|X1 is X1, itself a random variable, and the expected value of that is 1/2 (just the expected value of the marginal uniform distribution.

The second part, a little trickier for me, is the same idea, I'm finding the variance of the random variable X1 (which is the expected value of the conditional distribution X2, and that is 1/12. Then I add that to the expected value of the variance of the conditional distribution. That variable is X1^2, which is equal to the variance of X1 plus the mean of X1 squared.

Am I understanding this right?

Long story short, E(X2)=1/2, Var(X2)= 5/12 (this part is 1/12 + 1/12 + (1/2)^2

Sorry, I hope this is readable.b]

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# Homework Help: Expected value and variance of a conditional pdf (I think I have it)

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