Chapter 12

Assume that $$ \Omega=[1,2,3,4,5,6\\} $$ \(A=\\{1,3,5\\}\), and \(B=\\{1,2,3\\}\) Are \(A\) and \(B\) disjoint?

Let \(X\) be a continuous random variable with \(P(X>x)=e^{-a x}, \quad x \geq 0\) where \(a\) is a positive constant. Find \(E(X)\) and \(\operatorname{var}(X)\).

A family has two children. One of their children is a girl. Find the probability that both children are girls.

Automated chemical synthesis of DNA has made it possible to custom-order moderate-length DNA sequences from commercial suppliers. Assume that a single nucleotide weighs about \(5.6 \times 10^{-22}\) gram and that there are four kinds of nucleotides. If you wish to order all possible DNA sequences of a fixed length, at what length will your order exceed (a) \(100 \mathrm{~kg}\) and (b) the mass of the Earth \(\left(5.9736 \times 10^{24} \mathrm{~kg}\right)\) ?

Assume that a population consists of the three numbers 1 , 6, and \(8 .\) List all samples of size 2 that can be drawn from this population with replacement, and find the sample mean of each sample.

Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) are i.i.d. with $$ X_{i}=\left\\{\begin{aligned} -1 & \text { with probability } 0.2 \\ 1 & \text { with probability } 0.5 \\ 2 & \text { with probability } 0.3 \end{aligned}\right. $$ What can you say about \(\frac{1}{n} \sum_{i=1}^{n} X_{i}\) as \(n \rightarrow \infty ?\)

Suppose that the probability mass function of a discrete random variable \(X\) is given by the following table: $$ \begin{array}{rc} \hline \boldsymbol{x} & \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) \\ \hline-3 & 0.2 \\ -1 & 0.3 \\ 1.5 & 0.4 \\ 2 & 0.1 \\ \hline \end{array} $$ Find and graph the corresponding distribution function \(F(x)\).

Assume that \(\Omega=\\{1,2,3,4,5\\}\) \(P(1)=0.1, P(2)=0.2\), and \(P(3)=P(4)=0.05 .\) Furthermore, assume that \(A=\\{1,3,5\\}\) and \(B=\\{2,3,4\\}\) Find \(P(5)\).

Let \(X\) be a continuous random variable with density function \(f(x)=\left\\{\begin{array}{cl}(a-1) x^{-a} & \text { for } x>1 \\ 0 & \text { for } x \leq 1\end{array}\right.\) (a) Show that \(E(X)=\infty\) when \(a \leq 2\). (b) Compute \(E(X)\) when \(a>2\).

You roll two fair dice. Find the probability that the first die is a 4 given that the sum is 7 .

You plan a trip to Europe during which you wish to visit London, Paris, Amsterdam, Rome, and Heidelberg. Because you want to buy a railway ticket before you leave, you must decide on the order in which you will visit these five cities. How many different routes are there?

Use a graphing calculator to generate five samples, each of size 6, from a uniform distribution over the interval \((0,1) .\) Compute the sample means of each sample.

Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) are independent random variables with \(P\left(X_{i}>x\right)=e^{-2 x} .\) What can you say about \(\frac{1}{n} \sum_{i=1}^{n} X_{i}\) as \(n \rightarrow \infty ?\)

Suppose the probability mass function of a discrete random variable \(X\) is given by the following table: $$ \begin{array}{rl} \hline \boldsymbol{x} & \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) \\ \hline-1 & 0.2 \\ -0.5 & 0.25 \\ 0.1 & 0.1 \\ 0.5 & 0.1 \\ 1 & 0.35 \\ \hline \end{array} $$ Find and graph the corresponding distribution function \(F(x)\).

You roll two fair dice. Find the probability that the first die is a 5 given that the minimum of the two numbers is a 3 .

Five people line up for a photograph. How many different lineups are possible?

Let \(\left(X_{1}, X_{2}, \ldots, X_{n}\right)\) denote a sample of size \(n\). Show that $$ \sum_{k=1}^{n}\left(X_{k}-\bar{X}\right)=0 $$ where \(\bar{X}\) is the sample mean.

Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) are independent random variables with density function $$ f(x)=\frac{1}{\pi\left(1+x^{2}\right)}, \quad x \in \mathbf{R} $$ Can you apply the law of large numbers to \(\frac{1}{n} \sum_{i=1}^{n} X_{i} ?\) If so, what can you say about \(\frac{1}{n} \sum_{i=1}^{n} X_{i}\) as \(n \rightarrow \infty\) ?

Let \(X\) be a random variable with distribution function $$ F(x)=\left\\{\begin{array}{ll} 0 & x<-2 \\ 0.2 & -2 \leq x<0 \\ 0.3 & 0 \leq x<1 \\ 0.7 & 1 \leq x<2 \\ 1 & x \geq 2 \end{array}\right. $$ Determine the probability mass function of \(X\).

Assume that \(\Omega=\\{1,2,3,4,5\\}\) \(P(1)=0.1, P(2)=0.2\), and \(P(3)=P(4)=0.05 .\) Furthermore, assume that \(A=\\{1,3,5\\}\) and \(B=\\{2,3,4\\}\) Find \(P\left(A^{c}\right)\).

Denote by the density of a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) $$ f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^{2} / 2 \sigma^{2}} $$ for \(-\infty

You toss a fair coin three times. Find the probability that the first coin is heads given that at least one head occurred.

You have just bought seven different books. In how many ways can they be arranged on your bookshelf?

Let \(\left(X_{1}, X_{2}, \ldots, X_{n}\right)\) denote a sample of size \(n\). Show that $$ n \bar{X}^{2}=\frac{1}{n}\left(\sum_{k=1}^{n} X_{k}\right)^{2} $$ where \(\bar{X}\) is the sample mean.

Assume that \(\Omega=\\{1,2,3,4,5\\}\) \(P(1)=0.1, P(2)=0.2\), and \(P(3)=P(4)=0.05 .\) Furthermore, assume that \(A=\\{1,3,5\\}\) and \(B=\\{2,3,4\\}\) Find \(P(A \cup B)\)

Suppose that \(f(x)\) is the density function of a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). Show that $$ \mu=\int_{-\infty}^{\infty} x f(x) d x $$ is the mean of this distribution. (Hint: Use substitution.)

You toss a fair coin three times. Find the probability that at least two heads occurred given that the second toss resulted in heads.

Four cars arrive simultaneously at an intersection. Only one car can go through at a time. In how many different ways can they leave the intersection?

Assume that a sample of size \(n\) has \(l\) distinct values \(x_{1}, x_{2}, \ldots, x_{l}\), where \(x_{k}\) occurs \(f_{k}\) times in the sample. Explain why the sample mean is given by the formula $$ \bar{X}=\frac{1}{n} \sum_{k=1}^{l} x_{k} f_{k} $$

Let \(S=\\{1,2,3, \ldots, 10\\}\), and assume that $$ p(k)=\frac{k}{N}, k \in S $$ where \(N\) is a constant. (a) Determine \(N\) so that \(p(k), k \in S\), is a probability mass function. (b) Let \(X\) be a discrete random variable with \(P(X=k)=p(k)\). Find the probability that \(X\) is less than 8 .

You toss a fair coin four times. Find the probability that four heads occurred given that the first toss and the third toss resulted in heads.

How many four-letter "words" with no repeated letters can you form from the 26 letters of the alphabet?

Assume that a sample of size \(n\) has \(l\) distinct values \(x_{1}, x_{2}, \ldots, x_{l}\), where \(x_{k}\) occurs \(f_{k}\) times in the sample. Explain why the sample variance is given by the formula $$ S^{2}=\frac{1}{n-1}\left[\sum_{k=1}^{l} x_{k}^{2} f_{k}-\frac{1}{n}\left(\sum_{k=1}^{l} x_{k} f_{k}\right)^{2}\right] $$

Assume that \(E\left(e^{c X}\right)<\infty\) for \(c>0 .\) Use Markov's inequality to prove Bernstein's inequality, $$ P(X \geq x) \leq e^{-c x} E\left(e^{c X}\right) $$ for \(c>0\)

Geometric Distribution In Example 2, we tossed a coin repeatedly until the first heads showed up. Assume that the probability of heads is \(p\), where \(p \in(0,1)\). Let \(Y\) be a random variable that counts the number of trials until the first heads shows up. (a) Show that \(P(Y=1)=p, P(Y=2)=(1-p) p\), and \(P(Y=3)=(1-p)^{2} p\) (b) Explain why $$ P(Y=j)=(1-p)^{j-1} p $$ for \(j=1,2, \ldots\) This equation is called the geometric distribution. (c) Prove that $$ \sum_{j \geq 1} P(Y=j)=1 $$ as follows: (i) For \(0 \leq q<1\), define $$ S_{n}=1+q+q^{2}+\cdots+q^{n} $$ Show that $$ S_{n}-q S_{n}=1-q^{n+1} $$ and conclude from this equation that $$ S_{n}=\frac{1-q^{n+1}}{1-q} $$ (ii) Show that $$ P(Y \leq k)=\sum_{j=1}^{k} P(Y=j)=p \sum_{j=1}^{k}(1-p)^{j-1} $$ Use your results in (i) to show that this formula simplifies to $$ 1-(1-p)^{k} $$ and conclude from this equation that $$ \lim _{k \rightarrow \infty} P(Y \leq k)=1 $$ which is equivalent to $$ \sum_{j \geq 1} P(Y=j)=1 $$

You toss a fair coin four times. Find the probability of no more than three heads given that at least one toss resulted in heads.

A committee of 3 people must be chosen from a group of 10. The committee consists of a president, a vice president, and a treasurer. How many committees can be selected?

Assume that \(X\) is exponentially distributed with parameter \(\lambda=3.0 .\) (a) Assume that a sample of size 50 is taken from this population. What is the approximate distribution of the sample mean? (b) Assume now that 1000 samples, each of size 50 , are taken from this population and a histogram of the sample means of each of the samples is produced. What shape will the histogram be approximately?

15\. Toss a fair coin 400 times. Use the central limit theorem and the histogram correction to find an approximation for the probability of getting at most 190 heads.

The following table contains the number of leaves per basil plant in a sample of size 25 : \(\begin{array}{lllll}19 & 21 & 20 & 13 & 18 \\ 14 & 17 & 14 & 17 & 17 \\ 13 & 15 & 12 & 15 & 17 \\ 15 & 16 & 18 & 17 & 14 \\ 14 & 14 & 13 & 20 & 13\end{array}\) (a) Find the relative frequency distribution. (b) Compute the average value by (i) averaging the values in the table directly and (ii) using the relative frequency distribution obtained in (a).

Assume that a quantitative character is normally distributed with mean \(\mu\) and standard deviation \(\sigma .\) Determine what fraction of the population falls into the given interval. $$ [\mu, \infty) $$

Three different awards are to be given to a class of 15 students. Each student can receive at most one award. Count the number of ways these awards can be given out.

Assume that \(X\) is exponentially distributed with parameter \(\lambda=3.0 .\) Assume that a sample of size 50 is taken from this population and that the sample mean of this sample is calculated. How likely is it that the sample mean will exceed \(0.43\) ?

Toss a fair coin 150 times. Use the central limit theorem and the histogram correction to find an approximation for the probability that the number of heads is at least 70 .

The following table contains the number of aphids per plant in a sample of size 30 : \(\begin{array}{rrrrrr}15 & 27 & 13 & 2 & 0 & 16 \\ 26 & 0 & 2 & 1 & 17 & 15 \\\ 21 & 13 & 5 & 0 & 19 & 25 \\ 12 & 11 & 0 & 16 & 22 & 1 \\ 28 & 9 & 0 & 0 & 1 & 17\end{array}\) (a) Find the relative frequency distribution. (b) Compute the average value by (i) averaging the values in the table directly and (ii) using the relative frequency distribution obtained in (a).

Assume that a quantitative character is normally distributed with mean \(\mu\) and standard deviation \(\sigma .\) Determine what fraction of the population falls into the given interval. $$ [\mu-2 \sigma, \mu+\sigma] $$

Assume that \(P\left(A \cap B^{c}\right)=0.1, P\left(B \cap A^{c}\right)=0.5\), and \(P\left((A \cup B)^{c}\right)=0.2\). Find \(P(A \cap B)\).

A screening test for a disease shows a positive result in \(92 \%\) of all cases when the disease is actually present and in \(7 \%\) of all cases when it is not. Assume that the prevalence of the disease is 1 in 600 . If the test is administered to a randomly chosen individual, what is the probability that the result is positive?

You have just enough time to play 4 songs out of 10 stored on your phone. In how many ways can you program your phone to play the 4 songs?

Toss a fair coin 200 times. (a) Use the central limit theorem and the histogram correction to find an approximation for the probability that the number of heads is at least 120 . (b) Use Markov's inequality to find an estimate for the event in (a), and compare your estimate with that in (a).