I have $X_{1}$,$X_{2}$… $X_{n+1}$ independent uniform random variables on [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?
$E(Z|X_{n+1}<Z<c)$
I am aware that I should ideally find $F(Z|X_{n+1}<Z<c)$ in order to find the pdf and then the expectation but am quite stuck, I haven't been able to find solutions elsewhere.
Best Answer
Let $c > 0$. We have $f_X(x) = [0 < x < 1], \, f_Z(x) = n x^{n - 1} [0 < x < 1]$, $$\operatorname{E}(Z \mid X < Z < c) = \frac {\operatorname{E}(Z \, [X < Z < c])} {\operatorname{P}(X < Z < c)} = \\ \frac {\iint_{x < z < c} z f_X(x) f_Z(z) \, dx dz} {\iint_{x < z < c} f_X(x) f_Z(z) \, dx dz} = \frac {\int_0^{\min(c, 1)} z^{n + 1} dz} {\int_0^{\min(c, 1)} z^n dz}.$$