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I am trying to calculate a CDF of a random variable $x$ which has an upper bound $z$, which is itself a random variable with distribution $G(z)$ on some interval $[z1,z2]$.

E.g. $x \sim U[0,z]$ and $z \sim U[1,2]$.

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    @TAK Yes, you are right that $f(x,z)=\frac{1}{z}$ but be careful about their supports. Since 0 and 1, the actual joint density is f(x, z)=\frac{1}{z}\mathbb{I}[0.2012-10-03

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The conditional pdf of $X$ given $Z$ is $f(x|z)=\frac{1}{z}\mathbb{I}[0. The marginal pdf of $Z$ is $f(z)=\mathbb{I}[1. So the joint pdf of $X$ and $Z$ is $f(x,z)=f(x|z)f(z)=\frac{1}{z}\mathbb{I}[0 Then we can integrate over $z$ and get $f(x)=\int_{\max(x,1)}^{2} f(x,z)dz=\int_{\max(x,1)}^{2}\frac{dz}{z}=\log2-\log(\max(x,1))$

You can proceed by integrating it to get the CDF of $X$. You need to consider two different conditions when $x > 1$ and $x < 1$.

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    @TAK The accepting answer option should be somewhere on this page. If you can't find it, that's fine with me. Glad to help. I learn something from this as well.2012-10-16