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Let us define two, independent random variables $X, Y$. We know that: $$X - N(0, 1) \wedge Y-N(0,1).$$ Our task is to find characteristic function of a product of $X, Y$.
I know that:

  1. $\varphi_X(t) = e^{-\frac{t^2}{2}}$,
  2. $\varphi_{X+Y}(t) = \varphi_X(t) \varphi_Y(t)$.

Unfortunately I have no idea how to find $\varphi_{XY}$. I would appreciate any tips or hints.

Hendrra
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1 Answers1

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Hint: as a consequence of Fubini's theorem and the independence of $X$ and $Y$, $$ \mathbb E\left[f(X,Y)\right]=\int_{\mathbb R}\int_{\mathbb R}f(x,y)\mathrm dP_X(x)\mathrm dP_Y(y)=\int_{\mathbb R}\int_{\mathbb R}f(x,y) \mathrm dP_Y(y)\mathrm dP_X(x)=\int_{\mathbb R}\mathbb E\left[f(x,Y)\right] \mathrm dP_X(x). $$ for all measurable function $f\colon \mathbb R^2\to \mathbb R$. Apply this to $f\colon (x,y)\mapsto e^{itxy}$ and use the expression for the characteristic function of a standard normal random variable.

Davide Giraudo
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