Some applications

Let $X$ be the number of events in the sequence $A_{1}, A_{2}, \ldots , A_{n}$ which occur. Let $S_{m} = \mathbb {E}\left[\binom {X}{m}\right]$, the mean value of the random binomial coefficient $\binom {X}{m}$, and show that

Each person in a group of $n$ people chooses another at random. Find the probability:

Let $X$ have the binomial distribution with parameters $n$ and $p$, and show that

Find the limit of this expression as $n \rightarrow \infty$ and $p \rightarrow 0$, the limit being taken in such a way that $n p \rightarrow \lambda$ where $0 < \lambda < \infty$. Comment.

A coin is tossed repeatedly, and heads turns up with probability $p$ on each toss. Let $h_{n}$ be the probability of an even number of heads in the first $n$ tosses, with the convention that 0 is an even number. Find a difference equation for the $h_{n}$ and deduce that they have generating function $\frac{1}{2}\left\{ (1+2 p s-s)^{-1}+(1-s)^{-1}\right\}$.

An unfair coin is flipped repeatedly, where $\mathbb {P}\left(\mathrm{H}\right) = p = 1-q$. Let $X$ be the number of flips until HTH first appears, and $Y$ the number of flips until either HTH or THT appears. Show that $\mathbb {E}\left[s^{X}\right] = \left(p^{2} q s^{3}\right) /\left(1-s+p q s^{2}-p q^{2} s^{3}\right)$ and find $\mathbb {E}\left[s^{Y}\right]$.

The pile of (by now dog-eared) letters is dropped again and enveloped at random, yielding $X_{n}$ matches. Show that $\mathbb {P}\left(X_{n} = j\right) = (j+1) \mathbb {P}\left(X_{n+1} = j+1\right)$. Deduce that the derivatives of the $G_{n}(s) = \mathbb {E}\left[s^{X_{n}}\right]$ satisfy $G_{n+1}^{\prime } = G_{n}$, and hence derive the conclusion of Example (3.4.3), namely:

Let $X$ have a Poisson distribution with parameter $\Lambda$, where $\Lambda$ is exponential with parameter $\mu$. Show that $X$ has a geometric distribution.

Recall from Exercise (3.3.2) that each packet of an overpriced commodity contains a worthless plastic object. There are four types of object, and each packet is equally likely to contain any of the four. Let $T$ be the number of packets you open until you first have the complete set. Find $\mathbb {E}\left[s^{T}\right]$ and $\mathbb {P}\left(T = k\right)$.

A library permits a reader to hold at most $m$ books at any one time. Your holding before a visit is $B$ books, having the binomial $\operatorname {bin}(m, p)$ distribution. On each visit, a held book is retained with probability $r$, and returned otherwise; given the number $R$ of retained books, you borrow a further $N$ books with the $\operatorname {bin}(m-R, \alpha )$ distribution. Assuming the usual independence, find the distribution of the number $A = R+N$ of books held after a visit. Show that, after a large number of visits, the number of books held has approximately the $\operatorname {bin}(m, s)$ distribution where $s = \alpha /[1-r(1-\alpha )]$. You may assume that $0 < \alpha , p, r < 1$.

Matching yet again, Exercise (5.2.7) revisited. Each envelope is addressed correctly with probability $t$, and incorrectly otherwise (assume the usual independence). It is decided that an incorrectly addressed envelope cannot count as a match even when containing the correct letter. Show that the number $Y_{n}$ of correctly addressed matches satisfies

Note the Poisson limit distribution.

A random variable is called simple if it may take only finitely many values. Show that a simple random variable $X$ may be simulated using a finite sequence of Bernoulli random variables, in the sense that there exist independent Bernoulli random variables $\left\{ B_{j}: j = 1,2, \ldots , m\right\}$ with respective parameters $b_{j}$, and real numbers $\left\{ r_{j}\right\}$, such that $Y = \sum_{j} r_{j} B_{j}$ has the same distribution as $X$.