problem 28

4.2.4 problem 28

Let $I_{k}$ be the indicator variable for the $k$ -th location, so that $I_{k} = 1$ if $k$ -th location has a treasure and $I_{k} = 0$ otherwise.

Let $X$ be number of locations William checks to get $t$ treasures, and $X_{j} \sim HGeom (t - 1, n - 1 - (t - 1), j)$ be the number of treasures found within $j$ checked locations.

By symmetry. $P (I_{k} = 1) = \frac{t}{n}$ . Then,

P (X = k) = P (I_{k} = 1) P (X_{k - 1} = t - 1) = \frac{t}{n} \frac{(\frac{t - 1}{t - 1}) (\frac{n - 1 - t + 1}{k - 1 - t + 1})}{(\frac{n - 1}{k - 1})} = \frac{t}{n} \frac{(\frac{n - t}{k - t})}{(\frac{n - 1}{k - 1})}

E (X) = n \sum k = t kP (I_{k} = 1) P (X_{k - 1} = t - 1) = n \sum k = t (k \frac{t}{n} \frac{(\frac{n - t}{k - t})}{(\frac{n - 1}{k - 1})}) = \frac{(n + 1) t}{t + 1}

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