Expected Values, Covariance of two independent random variables

Question

Hello there can anyone helo me understand the question? The question noted that X and Z are independent but the equation is Y = X^2 + Z it would be a great help, thank you!

Answer 1

We have $(X,Z)\overset{\rm iid}\sim\mathcal N(0,1^2)$ (independent and standard normal distributed random variables).

Then we define $Y:=X^2+Z$.

So, clearly, when you encounter "$Y$", substitute "$X^2+Z$" and make use of the given properties for those independent and standard normal distributed random variables.

a) Show $\mathsf E(Y\mid X)=X^2$.

Seems obvious.

b) Show that $\mu_Y=1$

Use the Tower property; also known as the Law of Iterated Expectation.

c) Show that $\mathsf E(XY)=0$ and you are given a helpful hint.

Use the hint; it is truly helpful.

d) Show that $\mathsf {Cov}(X,Y)=0$ and thus that $\mathsf {Corr}(X,Y)=0$

Just use the definition of covariance (and correlation) and the results from the prior questions.