Problem source

\begin{questionparts} 
\item Let $X_{1}, X_{2}, \dots, X_{n}$
be independent
random variables each of which is
uniformly distributed on $[0,1]$.
Let $Y$ be the
largest of $X_{1}, X_{2}, \dots, X_{n}$. By using the fact
that $Y<\lambda$ if and only if $X_{j}<\lambda$ for
$1\leqslant j\leqslant n$, find the probability density function of $Y$.
Show that the variance of $Y$
is 
\[\frac{n}{(n+2)(n+1)^{2}}.\]
\item 
The probability that a neon light switched on at time $0$ will have failed
by a time $t>0$ is $1-\mathrm{e}^{-t/\lambda}$ where $\lambda>0$. I switch
on $n$ independent neon lights at time zero. Show that the expected time
until the first failure is $\lambda/n$.
\end{questionparts}

Solution source

\begin{questionparts}
\item $\,$ \begin{align*}
&& F_Y(\lambda) &= \mathbb{P}(Y < \lambda) \\
&&&= \prod_i \mathbb{P}(X_i < \lambda) \\
&&&= \lambda^n \\
\Rightarrow && f_Y(\lambda) &= \begin{cases} n \lambda^{n-1} & \text{if } 0 \leq \lambda \leq 1 \\ 0 & \text{otherwise} \end{cases} \\
\\
&& \E[Y] &= \int_0^1 \lambda f_Y(\lambda) \d \lambda \\
&&&= \int_0^1 n \lambda^n \d \lambda \\
&&&= \frac{n}{n+1} \\
&& \E[Y^2] &=  \int_0^1 \lambda^2 f_Y(\lambda) \d \lambda \\
&&&= \int_0^1 n \lambda^{n+1} \d \lambda \\
&&&= \frac{n}{n+2} \\
\Rightarrow && \var[Y] &= \E[Y^2]-(\E[Y])^2 \\
&&&= \frac{n}{n+2} - \frac{n^2}{(n+1)^2} \\
&&&= \frac{(n+1)^2n-n^2(n+2)}{(n+2)(n+1)^2} \\
&&&= \frac{n[(n^2+2n+1)-(n^2+2n)]}{(n+2)(n+1)^2} \\ 
&&&= \frac{n}{(n+2)(n+1)^2} 
\end{align*}

\item Using the same reasoning, we can see that

\begin{align*}
&& 1-F_Z(t) &= \mathbb{P}(\text{all lights still on after t}) \\
&&&= \prod_i e^{-t/\lambda} \\
&&&= e^{-nt/\lambda} \\
\\
\Rightarrow && F_Z(t) &= 1-e^{-nt/\lambda}
\end{align*}

Therefore $Z \sim Exp(\frac{n}{\lambda})$ and the time to first failure is $\lambda/n$

\end{questionparts}