Let Z ~ Exp(?), i.e., Z ? 0 and $f_Z(z) = ?e^{-?z}$, z ? 0. Conditioned on Z, let $X_1, X_2, ..., X_n$ be distributed independently as uniform in the range [0, Z], i.e., $X_i | Z$ are i.i.d., U[0, Z] random variables. Define the sequence $Y_n = max\{X_1, X_2, ..., X_n\}$. Does the sequence converge in the mean-square as $n ? ?$? If so, what is the limit? You need to provide a precise justification as to whether the sequence converges or not.
