Suppose $X$ and $I$ and independent, $X$ has a standard normal distribution and $I$ take values $1$ and $-1$ with equal probabilities.
Let $Y = IX$. How would I find the distribution of $Y$ and $X+Y$? and $E(Y|X)$, $E(Y^2|X)$?
Thanks so much!
Suppose $X$ and $I$ and independent, $X$ has a standard normal distribution and $I$ take values $1$ and $-1$ with equal probabilities.
Let $Y = IX$. How would I find the distribution of $Y$ and $X+Y$? and $E(Y|X)$, $E(Y^2|X)$?
Thanks so much!
If $U$ and $V$ are independent, and if $F$ is a positive (or bounded) measurable function on $\mathbb{R}^2$, then $E(F(U,V) | U) = G(U)$, where $G(u) = E(F(u,V))$.
It can help you to compute $E(Y|X)$ in your case. (for $E(Y^2|X)$, it's easy since $Y^2 = X^2$ which is $\sigma(X)$-measurable)
For the distribution of $Y$, just write the event $\{IX \in A\}$ as the disjoint union of the events $\{I = -1, X \in - A\}$ and $\{I = 1, X \in A\}$. Then use independence.
Same kind of reasoning for $X + Y = (1+I)X$.