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I have a random variable $X$, whose c.d.f. is known. Now I want to estimate the c.d.f of $Y = X \cdot Z$, where the only information I have about $Z$ is that it is a discrete r.v. which takes on three values.

What is the best estimate(in MMSE sense??) for the c.d.f. of $Y$.

My intuition says that it must be the c.d.f. of $X$, but I cant show it rigorously?

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  • $\begingroup$ With no further information about the distribution of $Z$, this is undoable. $\endgroup$ – Did Jan 29 '12 at 17:30
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Suppose $Z$ assumes one of three possible values $z_1$, $z_2$ or $z_3$ with certain probabilities: $$ \mathbb{P}(Z = z_k ) = p_k \quad k=1,2,3 $$ such that $p_1 + p_2 + p_3 = 1$. Since $X$ and $Z$ are independent, $$ F_Y(y) = \mathbb{P}(Y \leqslant y) = \mathbb{P}(z_1 X \leqslant y) p_1 + \mathbb{P}(z_2 X \leqslant y) p_2 + \mathbb{P}(z_3 X \leqslant y) p_3 $$ If we further assume that $z_1 > 0$ and $z_2>0$ and $z_3>0$ we get: $$ F_Y(y) = p_1 F_X\left(\frac{y}{z_1}\right) + p_2 F_X\left(\frac{y}{z_2}\right) + p_3 F_X\left(\frac{y}{z_3}\right) $$

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