My work:
I kind of know how to compute $E(Z)$ and $Var(Z)$ but still don't understand why $f_{Z|X}(Z|X)=1/X$ if $0<Z<X<1.$
Can anyone explain to me specifically how to come up with this?
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Welcome to MSE. Did you understand back in $Example.10.26$ why $f_{X|Y}(x) = \frac1y$? It's exactly the same here in $Exercise.10.41$. BTW, please do not use images for the critical part of the question, especially here your question is very specific and most of the info in the images is redundant (or can be easily typed). – Lee David Chung Lin Mar 09 '19 at 22:19
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In case you're wondering, the marginals of this process is the product of uniform distribution. One can also see this. In particular, here your $X$ is for $n = 2$ and $Z$ is for $n = 3$. – Lee David Chung Lin Mar 09 '19 at 22:32
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Small error: The $E(Z/X=x) = \int_{0}^{x} \frac{z}{x}dz= \frac{1}{2} x$ – Satish Ramanathan Mar 10 '19 at 06:49
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Another observation is $E(X)$ and $E(X^2)$ when evaluated with $f(x) = -ln(x) $ amounts to the same value as shown in example problem. So it is doubly correct. – Satish Ramanathan Mar 10 '19 at 06:53