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I have read Characteristic function of product of normal random variables But I really don't understand the derivation. Let X and Y be two independent random variables where X ~ N(0,1) and Y~exp(1) Form a new random variable Z = X*Y^(1/2) determine the characteristic function for Z and use this to determine $$ f_Z(z) $$

So starting by doing $$ \phi_{Z}(t) = E[e^{itz}] = E[e^{itX\sqrt{Y}}] = E[E[e^{itX\sqrt Y}|Y]] $$ After this I am lost. Why is this the same as $$ E[\phi_{X}(t\sqrt(y))] ? $$

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Maths
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  • Actually, the correct statement is that $$E[e^{itX\sqrt Y}|Y]=\phi_{X}(t\sqrt{Y})$$ and this holds because, for every $x$, $$E[e^{ixX}]=\phi_{X}(x)$$ and because $(X,Y)$ is independent. – Did Dec 22 '16 at 17:58
  • Could you please show a detailed calculation? Because I really don't get it. – Maths Dec 22 '16 at 18:02
  • I am afraid you will have to be much more specific (rather than posting duplicates on the site) before I can help you. What is it you do not get? What is $E(e^{itX\sqrt{Y}}\mid Y)$ according to you? – Did Dec 22 '16 at 18:07
  • According to me $E[e^{itX\sqrt(Y)}|Y]$ is the expected value of some function g =g(x,y). – Maths Dec 22 '16 at 18:21
  • No. Please check a rigorous definition of the conditional expectation of a random variable conditionally on (the sigma-algebra generated by) another random variable. – Did Dec 22 '16 at 18:30
  • When you say rigorous do you mean more rigorous than the one given by e.g. Gut? $$ E[Y|X=x] = \Sigma_y y*p_{Y|X=x (y)} $$ ? – Maths Dec 22 '16 at 18:41
  • Would you mean the formula $$E(Y\mid X=x)=\sum_yyp_{Y\mid X=x}(y)$$ by any chance? Which is not a definition of $E(X\mid Y)$, as you surely noted, and which assumes that $Y$ is discrete (not your case), and which requires to be able to identify $p_{Y\mid X=x}$ for every $x$... The last point in particular is odd since the object $p_{Y\mid X=x}$ is in general more involved than $E(Y\mid X)$. – Did Dec 22 '16 at 18:45
  • I do not understand what you are asking. So I will cease this thread. – Maths Dec 22 '16 at 18:59
  • Not asking anything, simply suggesting that you "check a rigorous definition of the conditional expectation of a random variable conditionally on (the sigma-algebra generated by) another random variable." – Did Dec 22 '16 at 19:44

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