Math Problems Vault

Problem 3.5.1

Each trial may result in any of $t$ given outcomes, the $i$ th outcome having probability $p_{i}$ . Let $N_{i}$ be the number of occurrences of the $i$ th outcome in $n$ independent trials. Show that

\mathbb {P}\left(N_{i} = n_{i} \text{ for } 1 \leq i \leq t\right) = \frac{n!}{n_{1}!n_{2}!\cdots n_{t}!} p_{1}^{n_{1}} p_{2}^{n_{2}} \cdots p_{t}^{n_{t}}

for any collection $n_{1}, n_{2}, \ldots , n_{t}$ of non-negative integers with sum $n$ . The vector $N$ is said to have the multinomial distribution.

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Problem 3.5.2

In your pocket is a random number $N$ of coins, where $N$ has the Poisson distribution with parameter $\lambda$ . You toss each coin once, with heads showing with probability $p$ each time. Show that the total number of heads has the Poisson distribution with parameter $\lambda p$ .

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Problem 3.5.3

Let $X$ be Poisson distributed where $\mathbb {P}\left(X = n\right) = p_{n}(\lambda ) = \lambda^{n} e^{-\lambda } / n$ ! for $n \geq 0$ . Show that $\mathbb {P}\left(X \leq k\right) = 1-\int_{0}^{\lambda } p_{k}(x) d x$ .

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Problem 3.5.4

A population of $b$ animals has had a number $a$ of its members captured, marked, and released. Let $X$ be the number of animals it is necessary to recapture (without re-release) in order to obtain $m$ marked animals. Show that

\mathbb {P}\left(X = n\right) = \frac{a}{b}\binom {a-1}{m-1}\binom {b-a}{n-m} /\binom {b-1}{n-1}

and find $\mathbb {E}\left[X\right]$ . This distribution has been called negative hypergeometric.

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Problem 3.5.5

Let $\Lambda$ be a positive random variable with density function $f$ and distribution function $F$ , and let $Y$ have the Poisson distribution with parameter $\Lambda$ . Show for $n = 0,1,2, \ldots$ that

\mathbb {P}\left(Y \leq n\right) = \int _{0}^{\infty } p_{n}(\lambda ) F(\lambda ) d \lambda , \quad \mathbb {P}\left(Y > n\right) = \int _{0}^{\infty } p_{n}(\lambda )[1-F(\lambda )] d \lambda

where $p_{n}(\lambda ) = e^{-\lambda } \lambda^{n} / n!$ .

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