Chapter 12

An urn contains 5 green and 25 blue balls. Balls are drawn at random, one at a time, until a green ball is selected. Each ball is replaced before the next ball is drawn. Let \(T\) denote the first time until a green ball is drawn. Find \(E(T)\) and \(\operatorname{var}(T)\).

Let \(N_{x}\) be the number of individuals that are still alive at age \(x\). Show that $$ -\ln \frac{N_{x+1}}{N_{x}} $$ can be estimated by $$ \int_{x}^{x+1} \lambda(u) d u $$ where \(\lambda(x)\) is the hazard-rate function at age \(x\).

An urn contains 10 green and 20 blue balls. Balls are drawn at random, one at a time, until a green ball is selected. Each ball is replaced before the next ball is drawn. Let \(T\) denote the first time until a green ball is drawn. Find \(E(T)\) and \(\operatorname{var}(T)\).

73\. An urn contains 1 black and 9 white balls. Balls are drawn at random until the black ball is selected. Find the probability that exactly 6 white balls will be drawn before the black one is if (a) each ball is replaced before the next ball is drawn and (b) balls are not replaced.

An urn contains 1 black and \(n-1\) white balls. Balls are drawn at random until the black ball is selected. Find the probability that exactly \(k\) white balls will be drawn before the black one is if (a) each ball is replaced before the next ball is drawn and (b) balls are not replaced.

Suppose the waiting time for the first success in an experiment is geometrically distributed with mean \(1 / p\). (a) Find the probability that the first success occurs on the \(k\) th trial. (b) The experiment is repeated after the first success. Assume that the waiting time for the second success has the same distribution as the waiting time for the first success. Find the probability mass function for the distribution of the second success.

A Bernoulli experiment with probability of success \(p\) is repeated until the \(n\) th success. Assume that each trial is independent of all others. Find the probability mass function of the distribution of the \(n\) th success. (This distribution is called the negative binomial distribution.)

Suppose \(X\) is Poisson distributed with parameter \(\lambda=2\). Find \(P(X=k)\) for \(k=0,1,2\), and \(3 .\)

Suppose \(X\) is Poisson distributed with parameter \(\lambda=0.5\). Find \(P(X=k)\) for \(k=0,1,2\), and 3 .

Suppose \(X\) is Poisson distributed with parameter \(\lambda=1\). (a) Find \(P(X \geq 2)\). (b) Find \(P(1 \leq X \leq 3)\).

Suppose \(X\) is Poisson distributed with parameter \(\lambda=0.2\). (a) Find \(P(X<3)\). (b) Find \(P(2 \leq X \leq 4)\).

Suppose \(X\) is Poisson distributed with parameter \(\lambda=1.5\). Find the probability that \(X\) exceeds \(3 .\)

Suppose \(X\) is Poisson distributed with parameter \(\lambda=1.2\). Find the probability that \(X\) is at most 3 .

Suppose \(X\) is Poisson distributed with parameter \(\lambda=2\). Find the probability that \(X\) is at least \(2 .\)

Suppose \(X\) is Poisson distributed with parameter \(\lambda=0.6\). Find the probability that \(X\) is less than \(3 .\)

Suppose the number of phone calls arriving at a switchboard per hour is Poisson distributed with mean 7 calls per hour. Find the probability that no phone calls arrive during a certain hour.

Suppose the number of phone calls arriving at a switchboard per hour is Poisson distributed with mean 3 calls per hour. (a) Find the probability that at least one phone call arrives between noon and \(1 \mathrm{P.M}\). (b) Assuming that phone calls in different hours are independent of each other, find the probability that no phone calls arrive between noon and 2 P.M.

Suppose the number of typos on a book page is Poisson distributed with mean \(0.5\). Find the probability that there is at least one typo on a given page.

Suppose the number of typos on a book page is Poisson distributed with mean \(0.1\). (a) Find the probability that there are no typos on a page. (b) How many pages with typos do you expect in a 200 -page book?

The number of amino acid substitutions in a given sequence is Poisson distributed with mean 3 . What is the probability of at least two substitutions?

The number of amino acid substitutions in a given sequence is Poisson distributed with mean 2. Given that there are substitutions on the sequence, what is the probability that there are at least two substitutions?

\(X\) and \(Y\) are independent and Poisson with mean 3 . (a) Find \(P(X+Y=2)\). (b) Given that \(X+Y=2\), find the probability that \(X=k\) for \(k=0,1\), and \(2 .\)

\(X\) is Poisson distributed with mean 2 , and \(Y\) is Poisson distributed with mean 3 . (a) Find \(P(X+Y=4)\) (b) Given that \(X+Y=1\), find the probability that \(X=1\).

Let \(X\) and \(Y\) be independent Poisson distributed variables with means 4 and 2 respectively. Calculate \(P(X=2 \mid X+Y=3)\).

Suppose \(X\) and \(Y\) are independent and Poisson with mean \(\lambda\). Given that \(X+Y=n\), find the probability that \(X=k\) for \(k=0,1,2, \ldots, n\)

Use the Poisson approximation. For a certain vaccine, 1 in 1000 individuals experiences some side effects. Find the probability that, in a group of 500 people, nobody experiences side effects.

Use the Poisson approximation. For a certain vaccine, 1 in 500 individuals experiences some side effects. Find the probability that, in a group of 200 people, at least 1 person experiences side effects.

Use the Poisson approximation. Down Syndrome About 1 in 700 births in the United States is affected by Down syndrome, a chromosomal disorder. Find the probability that there is at most 1 case of Down syndrome among 1000 births by (a) computing the exact probability and (b) using a Poisson approximation.

Use the Poisson approximation. Fragile \(X\) Syndrome About 1 in 1000 boys is affected by fragile \(X\) syndrome, a genetic disorder that causes learning difficulties. Find the probability that, in a group of 500 boys, nobody is affected by this disorder by (a) computing the exact probability and (b) using a Poisson approximation.