Moment generating function of the maximum of n independent uniform random variables.

Question

Let $(U_n)_n$ be a sequence of i.i.d random variables such that $U_{n} \sim U[0,1] \forall n$. Define $Y_{n}=\max\{U_{1}, \frac{U_{2}}{2}, \ldots, \frac{U_{n}}{n}\}$. How can I calculate the moment generating function for $Y_n$ directly? I understand that maybe one way is using law of total expectation.

Answer 1

Hint:

Find the CDF $F_Y(y)$: $$F_Y(y) := \mathbb{P}(Y \le y)=\prod_{k=1}^n \mathbb{P}(U_k\le yk)= n! \prod_{k=1}^n \min\left\{y,\frac{1}{k}\right\}$$ Compute the MGF $M_Y(t)$ from $F_Y(y)$: $$M_Y(t) = \int_0^1 e^{ty}F'_Y(y)dy=e^t -t \int_0^1 \left(e^{ty}n! \prod_{k=1}^n \min\left\{y,\frac{1}{k}\right\}\right)dy$$ The integral is the sum of $n$ integral $\int_{\frac{i-1}{n}}^{\frac{i}{n}} \left(e^{ty}n! \prod_{k=1}^n \min\left\{y,\frac{1}{k}\right\}\right)dy$ for $i = 1,...,n$ that can be calculated easily.