(Solved): (3) Suppose that we draw a random sample X1,X2,,Xn from a uniform distribution Unif(0, ...

(3) Suppose that we draw a random sample $X_1,X_2,\dots ,X_n$ from a uniform distribution $\mathrm{Unif}(0,?)$, but that $?$ is unknown; we want to build an estimator for $?$. (a) Suppose we're only allowed to draw one sample; show that $S=2X_1$ is an unbiased estimator for $?$. (b) Define $T=\max \left\{X_1,X_2,\dots ,X_n\right\}$. Show that $E[T]=\frac{n}{n+1}?$ for all $n$. 1 MATH 032 - FALL 2023 WEEKLY HOMEWORK 8 (c) Explain why $T$ is a biased estimator for $?$, but that the bias of $T$ tends to zero as $n$ tends to infinity. Does it match your intuition that $E[T]<?$ for every $n$ ? Hint: The first boxed formula on page 109 might be useful when you're computing the expectation of $T$ in part (b). You can use it to show that the CDF of $T$ is $(x/?{)}^n$ when $0?x??$.