Question
Consider an urri containing $n$ balls, numbered $1, \ldots, n$, and suppose that $k$ of them are randomly withdrawn. Let $X_{i}$ equal 1 if ball numbered $i$ is removed and let it be 0 otherwise. Show that $X_{1}, \ldots, X_{n}$ are exchangeable.
Step 1
$Y_{i}$ is the number of balls one must draw to obtain the $i$-th special ball, and $Y_{i}$ does not include the selection number of the first special ball drawn. Show more…
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Consider an urn containing $n$ balls numbered $1, \ldots, n,$ and suppose that $k$ of them are randomly withdrawn. Let $X_{i}$ equal 1 if ball number $i$ is removed and let $X_{i}$ be 0 otherwise. Show that $X_{1}, \ldots, X_{n}$ are exchangeable.
Suppose that balls are randomly removed from an um initially containing $n$ white and $m$ black balls. It was shown in Example $2 \mathrm{~m}$ that $E[X]=$ $1+m /(n+1)$, when $X$ is the number of draws needed to obtain a white ball. (a) Compute $\operatorname{Var}(X)$. (b) Show that the expected number of balls that need be drawn to amass a total of $k$ white balls is $k[1+m /(n+1)]$. HINT: Let $Y_{h}, i=1, \ldots, n+1$, denote the number of black balls withdrawn after the $(i-1)$ st white ball and before the $i$ th white ball. Argue that the $Y_{i}, i=1, \ldots, n+1$, are identically distributed.
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