Two limit theorems

Prove that, for $x \geq 0$, as $n \rightarrow \infty$,

It is well known that infants born to mothers who smoke tend to be small and prone to a range of ailments. It is conjectured that also they look abnormal. Nurses were shown selections of photographs of babies, one half of whom had smokers as mothers; the nurses were asked to judge from a baby's appearance whether or not the mother smoked. In 1500 trials the correct answer was given 910 times. Is the conjecture plausible? If so, why?

Let $X$ have the $\Gamma (1, s)$ distribution; given that $X = x$, let $Y$ have the Poisson distribution with parameter $x$. Find the characteristic function of $Y$, and show that

Explain the connection with the central limit theorem.

Let $X_{1}, X_{2}, \ldots$ be independent random variables taking values in the positive integers, whose common distribution is non-arithmetic, in that $\operatorname {gcd}\left\{ n: \mathbb {P}\left(X_{1} = n\right) > 0\right\} = 1$. Prove that, for all integers $x$, there exist non-negative integers $r = r(x), s = s(x)$, such that

Prove the local central limit theorem for sums of random variables taking integer values. You may assume for simplicity that the summands have span $1$, in that $\operatorname {gcd}\left\{ \left|x\right|: \mathbb {P}\left(X = x\right) > 0\right\} = 1$.

Let $X_{1}, X_{2}, \ldots$ be independent random variables having common density function $f(x) = 1 /\left\{ 2\left|x\right|(\log \left|x\right|)^{2}\right\}$ for $\left|x\right| < e^{-1}$. Show that the $X_{i}$ have zero mean and finite variance, and that the density function $f_{n}$ of $X_{1}+X_{2}+\cdots +X_{n}$ satisfies $f_{n}(x) \rightarrow \infty$ as $x \rightarrow 0$. Deduce that the $X_{i}$ do not satisfy the local limit theorem.

Let $X$ have the density function $f(x) = \sqrt{2 \pi x^{-3}} \exp \left(-\left\{ 2 x\right\}^{-1}\right), x > 0$. Show that $\phi (is) = \mathbb {E}\left[e^{-s X}\right] = e^{-\sqrt{2 s}}, s > 0$, and deduce that $X$ has characteristic function

Let $\left\{ X_{r}: r \geq 1\right\}$ be independent with the distribution of the preceding Exercise (5.10.7). Let $U_{n} = n^{-1} \sum_{r = 1}^{n} X_{r}$, and $T_{n} = n^{-1} U_{n}$. Show that:

A sequence of biased coins is flipped; the chance that the $r$ th coin shows a head is $\Theta_{r}$, where $\Theta_{r}$ is a random variable taking values in $(0,1)$. Let $X_{n}$ be the number of heads after $n$ flips. Does $X_{n}$ obey the central limit theorem when:

In an election with two candidates, each of the $v$ voters is equally likely to vote for either candidate, and they vote independently of one another. Show that, when $v$ is large, the probability that the winner was ahead when $\lambda v$ votes had been counted (where $\lambda \in (0,1)$) is approximately $\frac{1}{2}+(1 / \pi ) \sin^{-1} \sqrt{\lambda }$.

By considering the central limit theorem for the sum of independent Poisson-distributed random variables, show that

Let $f$ be a mass function on $\left\{ 1,2, \ldots \right\}$ with finite mean $\mu$. Consider a large population of households with independent sizes distributed as $f$. An individual A is selected uniformly at random from the entire population. Show that the probability A belongs to a household of size $x$ is approximately $x f(x) / \mu$. Find the probability generating function of this distribution in terms of that of $f$.

Let $X_{1}, X_{2}, \ldots$ be independent, strictly positive continuous random variables with common density function $f_{X}$ and finite mean, and let $Z_{r} = X_{r} / S_{n}$ where $S_{k} = \sum_{r = 1}^{k} X_{r}$. The $Z_{r}$ give rise to a partition of $(0,1]$ into the intervals $I_{r} = \left(S_{r-1}, S_{r}\right] / S_{n}$ for $r = 1,2, \ldots , n$. Let $U$ be uniform on $(0,1]$ and independent of the $X_{r}$, and let $L$ be the length of the interval into which $U$ falls. Show that $L$ has density function $f_{L}(z) = (z / \mathbb {E}\left[Z\right]) f_{Z}(z)$, where $Z = Z_{1}$. Find the characteristic function of $L$ in terms of that of $Z$.

Let $S_{n} = \sum_{r = 1}^{n} X_{r}$ be a continuous, symmetric random walk on $\mathbb {R}$ started at $0$, whose jumps are independent $N(0,1)$ random variables $X_{i}$. Define the area process $A_{n} = \sum_{m = 1}^{n} S_{m}$. Let $I_{m} = \left\{ -1 < A_{m} < 1\right\}$, and show that, with probability $1$, only finitely many of the $I_{m}$ occur.

At Bell Labs in the early 1950s, David Hagelbarger and Claude Shannon built machines to predict whether a human coin-flipper would call heads or tails. Hagelbarger's machine (a "sequence extrapolating robot", or "SEER") was correct on 5218 trials of 9795, and Shannon's machine (a "mind-reading (?) machine") was correct on 5010 trials of 8517. In each case, what is the probability of doing at least as well by chance?