Chapter 12

Suppose \(X\) is a random variable with mean \(-5\) and variance \(2 .\) What can you say about the probability that \(X\) deviates from its mean by at least \(4 ?\)

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

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 \(a>2\).

In Problems \(9-12\), 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 \(P(5)\)

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?

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

Suppose that \(X\) is a continuous random variable that takes on only nonnegative values. Set $$ G(x)=P(X>x) $$ (a) Show that $$ G^{\prime}(x)=-f(x) $$ where \(f(x)\) is the corresponding density function. (b) Assume that $$ \lim _{x \rightarrow \infty} x G(x)=0 $$ and use integration by parts and (a) to show that $$ E(X)=\int_{0}^{\infty} G(x) d x $$ (c) Let \(X\) be a continuous random variable with $$ P(X>x)=e^{-a x}, \quad x>0 $$ where \(a\) is a positive constant. Use \((12.35)\) to find \(E(X)\). (If you did Problem 8 , compare your answers.)

10\. 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?

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 \(n \rightarrow \infty ?\)

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 \(X\).

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

11\. 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 $$ \sum_{k=1}^{n}\left(X_{k}-\bar{X}\right)=0 $$ 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 $$F(x)=\left\\{\begin{array}{ll} 0 & x<0 \\ 0.05 & 0 \leq x<1.3 \\ 0.30 & 1.3 \leq x<1.7 \\ 0.85 & 1.7 \leq x<1.9 \\ 0.90 & 1.9 \leq x<2 \\ 1.0 & x \geq 2 \end{array}\right.$$ Determine the probability mass function of \(X\).

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.)

In Problems \(9-12\), 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 \(P(A \cup B)\).

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?

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.

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 .

In Problems 13-15, assume that $$ \Omega=\\{1,2,3,4\\} $$ and \(P(1)=0.1 .\) Furthermore, assume that \(A=\\{2,3\\}\) and \(B=\) \\{3. 4\\}. \(P(A)=0.7\), and \(P(B)=0.5\). Find \(P(3)\).

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 mean is given by the formula $$ \bar{X}=\frac{1}{n} \sum_{k=1}^{l} x_{k} f_{k} $$

A certain study showed that less than \(5 \%\) of the population suffers from a certain disorder. To get a more accurate estimate of this proportion, you plan to conduct another study. What sample size should you choose if you want to be at least \(95 \%\) sure that your estimate is within \(0.05\) of the true value?

Suppose a quantitative character is normally distributed with mean \(\mu=15.4\) and standard deviation \(\sigma=3.1 .\) Find an interval centered at the mean such that \(95 \%\) of the population falls into this interval. Do the same for \(99 \%\) of the population.

In Problems 13-15, assume that $$ \Omega=\\{1,2,3,4\\} $$ and \(P(1)=0.1 .\) Furthermore, assume that \(A=\\{2,3\\}\) and \(B=\) \\{3. 4\\}. \(P(A)=0.7\), and \(P(B)=0.5\). Set \(C=\\{1,2\\}\). Find \(P(C)\).

You toss a fair coin four times. Find the proability 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 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 $$ S^{2}=\frac{1}{n-1}\left[\sum_{k=1}^{I} x_{k}^{2} f_{k}-\frac{1}{n}\left(\sum_{k=1}^{l} x_{k} f_{k}\right)^{2}\right] $$

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 \(c>0\).

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).

In Problems 15-20, 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. \([\mu, \infty)\)

In Problems 13-15, assume that $$ \Omega=\\{1,2,3,4\\} $$ and \(P(1)=0.1 .\) Furthermore, assume that \(A=\\{2,3\\}\) and \(B=\) \\{3. 4\\}. \(P(A)=0.7\), and \(P(B)=0.5\). Find \(P\left((A \cap B)^{c}\right.\)

A screening test for a disease shows a positive test result in \(90 \%\) of all cases when the disease is actually present and in \(15 \%\) of all cases when it is not. Assume that the prevalence of the disease is 1 in 100 . If the test is administered to a randomly chosen individual, what is the probability that the result is negative?

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 \(\lambda=3.0\). (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?

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 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 \(P\left(A \cap B^{c}\right)=0.1, P\left(B \cap A^{c}\right)=0.5\), 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 from your favorite CD. In how many ways can you program your CD player to play the 4 songs?

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 \(0.43 ?\)