Chapter 9: Central Limit TheoremContent check — Probability (Grinstead and Snell), Chapter 9: Central Limit Theorem

Let $S_{100}$ be the number of heads that turn up in 100 tosses of a fair coin. Use the Central Limit Theorem to estimate

(a) $P(S_{100}\le 45)$⁠.

(b) $P(45<S_{100}<55)$⁠.

(c) $P(S_{100}>63)$⁠.

(d) $P(S_{100}<57)$⁠.

A true-false examination has 48 questions. June has probability 3/4 of answering a question correctly. April just guesses on each question. A passing score is 30 or more correct answers. Compare the probability that June passes the exam with the probability that April passes it.

A rookie is brought to a baseball club on the assumption that he will have a .300 batting average. (Batting average is the ratio of the number of hits to the number of times at bat.) In the first year, he comes to bat 300 times and his batting average is .267. Assume that his at bats can be considered Bernoulli trials with probability .3 for success. Could such a low average be considered just bad luck or should he be sent back to the minor leagues? Comment on the assumption of Bernoulli trials in this situation.

Assume that, as in Example 9.3, Dartmouth admits 1750 students. What is the probability of too many acceptances?

Let $S_n$ be the number of successes in $n$ Bernoulli trials with probability .8 for success on each trial. Let $A_n=S_n/n$ be the average number of successes. In each case give the value for the limit, and give a reason for your answer.

(a) ${\lim }_{n\to \infty }P(A_n=.8)$⁠.

(b) ${\lim }_{n\to \infty }P(.7n<$ $S_n<$ $.9n)$⁠.

(c) ${\lim }_{n\to \infty }P(S_n<.8n+.8\sqrt{n})$⁠.

(d) ${\lim }_{n\to \infty }P(.79<$ $A_n<$ $.81)$⁠.

A noodle machine in Spumoni’s spaghetti factory makes about 5 percent defective noodles even when properly adjusted. The noodles are then packed in crates containing 1900 noodles each. A crate is examined and found to contain 115 defective noodles. What is the approximate probability of finding at least this many defective noodles if the machine is properly adjusted?

In an opinion poll it is assumed that an unknown proportion $p$ of the people are in favor of a proposed new law and a proportion $1-p$ are against it. A sample of $n$ people is taken to obtain their opinion. The proportion $\bar{p}$ in favor in the sample is taken as an estimate of $p$⁠. Using the Central Limit Theorem, determine how large a sample will ensure that the estimate will, with probability .95, be correct to within .01.

Let $X$ be a continuous random variable with mean $\mu (X)$ and variance ${\sigma }^2(X)$⁠, and let $X^{\ast }=(X-\mu )/\sigma$ be its standardized version. Verify directly that $\mu (X^{\ast })=0$ and ${\sigma }^2(X^{\ast })=1$⁠.

Let $\{X_k\}$⁠, $1\le k\le n$⁠, be a sequence of random variables, all with mean $\mu$ and variance ${\sigma }^2$⁠, and $Y_k=X_k^{\ast }$ be their standardized versions. Let $S_n$ and $T_n$ be the sum of the $X_k$ and $Y_k$⁠, and $S_n^{\ast }$ and $T_n^{\ast }$ their standardized version. Show that $S_n^{\ast }=T_n^{\ast }=T_n/\sqrt{n}$⁠.

The price of one share of stock in the Pilsdorff Beer Company (see Exercise 8.2.12) is given by $Y_n$ on the $n$⁠th day of the year. Finn observes that the differences $X_n=Y_{n+1}-Y_n$ appear to be independent random variables with a common distribution having mean $\mu =0$ and variance ${\sigma }^2=1/4$⁠. If $Y_1=100$⁠, estimate the probability that $Y_{365}$ is

(a) $\ge 100$⁠.

(b) $\ge 110$⁠.

(c) $\ge 120$⁠.

Physicists say that particles in a long tube are constantly moving back and forth along the tube, each with a velocity $V_k$ (in cm/sec) at any given moment that is normally distributed, with mean $\mu =0$ and variance ${\sigma }^2=1$⁠. Suppose there are ${10}^{20}$ particles in the tube.

(a) Find the mean and variance of the average velocity of the particles.

(b) What is the probability that the average velocity is $\ge {10}^{-9}$⁠ cm/sec?