Let $X$, $Y$ be two independent real-valued random variables on $(\Omega,\mathcal{F},\mathbb{P})$. Suppose that $Y$ has a standard normal distribution.
a) show that the following properties are equivalent:
i) $\mathbb{E}[e^{X^2/2}]<\infty$
ii) $\mathbb{E}[e^{XY}]<\infty$
iii) $\mathbb{E}[e^{|XY|}]<\infty$
b) Show that if $\mathbb{E}[e^{X^2/2}]<\infty$ then $\mathbb{E}[e^{XY}|X]\geq1$ almost surely.
I do not know if $X$ has a density function, so I do not know how to argue using integrals. And I also not sure how to calculate the expectation for the product of two differently distributed variables.
Best Answer
As a consequence of Fubini's theorem and the independence of $X$ and $Y$, $$ \mathbb E\left[f(X,Y)\right]=\int_{\mathbb R}\int_{\mathbb R}f(x,y)\mathrm dP_X(x)\mathrm dP_Y(y)=\int_{\mathbb R}\int_{\mathbb R}f(x,y) \mathrm dP_Y(y)\mathrm dP_X(x)=\int_{\mathbb R}\mathbb E\left[f(x,Y)\right] \mathrm dP_X(x). $$ Use the fact that $Y$ has a normal distribution to evaluate for a fixed $x$ the quantity $\mathbb E\left[f(x,Y)\right]$ in the cases $f(x,y)=e^{xy}$ and $f(x,y)=e^{\left\lvert xy\right\rvert}$.