This is an extension of the previous question here.
Conditional expected value of a maximum of uniform random variables
I have $X_{1}$,$X_{2}$… $X_{n}$ independent uniform random variables on $[0,1]$.
I also have $Y_{n+1}$ which is a uniform distribution on $[0,a]$
where $a\in [0,1]$
let $Z=max(X_{1},X_{2}… X_{n})$
let $c$ be a constant s.t $c \in [0,1]$
What is the following conditional expectation?
$E(Z|Y_{n+1}<Z<c)$
The previous post contains a solution for when $a=1$ but I am unsure how to proceed when the supports are different.
Best Answer
$$\iint_{y<z<c}f_Y(y)f_Z(z)dydz=\iint_Dnz^{n-1}dydz$$ where $D=\{(z,y): 0<y<a \wedge 0<z<1 \wedge y<z<c\}$
It is best to do this with dz as the inner integral.
$$\int_0^{min(a,c)}\int_y^c nz^{n-1}dzdy=\int_0^{min(a,c)} c^n-y^n=min(a,c)c^n-\frac{min(a,c)^{n+1}}{n+1}$$
Therefore the expectation becomes
$$E(Z|Y_{n+1}<Z<c)=\frac{\iint_Dnz^{n}dydz}{\iint_Dnz^{n-1}dzdy}=\frac{\int_0^{min(a,c)} \frac{n}{n+1}(c^{n+1}-y^{n+1}dy)}{\int_0^{min(a,c)} c^n-y^ndy}$$ $$=\frac{min(a,c)\frac{n}{n+1}c^{n+1}-\frac{n}{(n+1)(n+2)}min(a,c)^{n+2}}{min(a,c)c^n-\frac{min(a,c)^{n+1}}{n+1}}$$ which is the same as the previous result whenever $min(a,c)=c$
Much thanks to Maxim for the hints.