I'm struggling to solve the following problem from Resnick's book:
Suppose $X_{1}$,$X_{2}$ are iid random variables with common N(0,1) distribution. Define:
$Y_{n} = \frac{X_{1}}{\frac{1}{n}+|X_{2}|}$.
Use Fubini's theorem to verify that
$E(Y_{n}) = 0$.
Note as n $\rightarrow \infty$, $Y_{n} \rightarrow Y=\frac{X_{1}}{|X_{2}|}$
and the expectation of Y doest not exist.
I've tried to attack the problem doing the following:
$E(Y_{n}) = \int_{\Omega_{Y_{n}}}Y_{n}dP = \int_{\Omega_{2}} \int_{\Omega_{1}}\frac{X_{1}}{\frac{1}{n}+|X_{2}|} dP_{1}dP_{2} = \int_{\Omega_{2}}\frac{1}{\frac{1}{n}+|X_{2}|} [\int_{\Omega_{1}} X_{1}dP_{1}]dP_{2} = \int_{\Omega_{2}}\frac{1}{\frac{1}{n}+|X_{2}|} E[X_{1}]dP_{2} = E[X_{1}]\int_{\Omega_{2}}\frac{1}{\frac{1}{n}+|X_{2}|}dP_{2} $
And then I got stuck at this part, since I don't know how to treat this Lebesgue integral.
Had the same problem on finding E(Y), where $Y=\frac{X_{1}}{|X_{2}|}$
$E(Y) =E[X_{1}]\int_{\Omega_{2}}\frac{1}{|X_{2}|}dP_{2} $
How to solve this problem?
Why expectation of $Y_{n}$ exists, but not Y?
Thanks
You got $\mathbb E\left[X_n\right]=\mathbb E\left[X_1\right]\int_{\Omega_{2}}\frac{1}{\frac 1n+|X_{2}|}dP_{2}$. Since $\int_{\Omega_{2}}\frac{1}{\frac 1n+|X_{2}|}dP_{2}$ is finite for any fixed $n$ (the integrand is smaller than $n$), we get that $\mathbb E\left[X_n\right]=0$.
The expectation of $Y$ does not exist since $\mathbb E\left[\left\lvert X_1\right\rvert/\left\lvert X_2\right\rvert\right]$ is infinite. To see this, use Fubini's theorem to be reduced to prove that $\mathbb E\left[1/\left\lvert X_2\right\rvert\right]$ is infinite.