problem 42

4.3.5 problem 42

Let $I_{j}$ be the indicator variable for the $j$ -th toy being of a new type. The number of toy types after collecting $t$ toys is $X = \sum_{j = 1}^{t} I_{j}$ . $P (I_{j} = 1) = {(\frac{n - 1}{n})}^{j - 1}$ . Thus,

E (X) = E (t \sum j = 1 I_{j}) = t \sum j = 1 E (I_{j}) = t \sum j = 1 {(\frac{n - 1}{n})}^{j - 1} = n - n {(\frac{n - 1}{n})}^{t}

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