Thursday, October 23, 2008

Distribute n distinguishable balls into r boxes

Suppose $n$ distinguishable balls are randomly distributed into $r$ boxes. If $S_r$ is the number of empty boxes, $S_r = X_1 + ... +X_r$ where $X_i$ is 1 or 0 according to whether box $i$ is empty or not, $1<=i<=r$.

Calculate:
(1) E[X_i]
(2) E[X_iX_j]
(3) E[S_r] and Var[S_r]

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