Chapter 12

Toss a fair coin twice. Let \(X\) be the random variable that counts the number of tails in each outcome. Find the probability mass function describing the distribution of \(X\).

Show that $$ f(x)=\left\\{\begin{array}{cc} 3 e^{-3 x} & \text { for } x>0 \\ 0 & \text { for } x \leq 0 \end{array}\right. $$ is a density function. Find the corresponding distribution function.

In Problems \(1-4\), determine the sample space for each random experiment. The random experiment consisting of tossing a coin three times.

Suppose you draw 2 cards from a standard deck of 52 cards. Find the probability that the second card is a spade given that the first card is a club.

Suppose that you want to investigate the influence of light and fertilizer levels on plant performance. You plan to use five fertilizer and two light levels. For each combination of fertilizer and light level, you want four replicates. What is the total number of replicates?

The following data represent the number of aphids per plant found in a sample of 10 plants: $$ 17,13,21,47,3,6,12,25,0,18 $$ Find the median, the sample mean, and the sample variance.

Let \(X\) be exponentially distributed with parameter \(\lambda=1 / 2\). Use Markov's inequality to estimate \(P(X \geq 3)\), and compare your estimate with the exact answer.

Toss a fair coin four times. Let \(X\) be the random variable that counts the number of heads. Find the probability mass function describing the distribution of \(X\).

Show that $$ f(x)=\left\\{\begin{array}{ll} \frac{1}{2} & \text { for } 0

In Problems \(1-4\), determine the sample space for each random experiment. The random experiment consisting of rolling a six-sided die twice.

Suppose you draw 2 cards from a standard deck of 52 cards. Find the probability that the second card is a spade given that the first card is a spade.

Suppose that you want to investigate the effects of leaf damage on the performance of drought-stressed plants. You plan to use three levels of leaf damage and four different watering protocols. For each combination of leaf damage and watering protocol, you plan to have three replicates. What is the total number of replicates?

The following data represent the number of seeds per flower head in a sample of nine flowering plants: $$ 27,39,42,18,21,33,45,37,21 $$ Find the median, the sample mean, and the sample variance.

Roll a fair die twice. Let \(X\) be the random variable that gives the absolute value of the differences betwen the two numbers. Find the probability mass function describing the distribution of \(X\).

Determine \(c\) such that $$ f(x)=\frac{c}{1+x^{2}}, \quad x \in \mathbf{R} $$ is a density function.

In Problems \(1-4\), determine the sample space for each random experiment. An urn contains five balls numbered \(1-5\), respectively. The random experiment consists of selecting two balls simultaneously without replacement.

Suppose you draw 3 cards from a standard deck of 52 cards. Find the probability that the third card is a club given that the first two cards are spades.

Coleomegilla maculata, a lady beetle, is an important predator of egg masses of Ostrinia nubialis, the European corn borer. \(C\). maculata also feeds on aphids and maize pollen. To study its food preferences, you choose two satiation levels for \(C\). maculata and combinations of two of the three food sources (i.e., either egg masses and aphids, egg masses and pollen, or aphids and pollen). For each experimental protocol, you want 20 replicates. What is the total number of replicates?

The following data represent the age of patients in a clinical trial: $$ 28,45,34,36,30,42,35,45,38,27 $$ Find the median, the sample mean, and the sample variance.

Roll a fair die twice. Let \(X\) be the random variable that gives the maximum of the two numbers. Find the probability mass function describing the distribution of \(X\).

Determine \(c\) such that $$ f(x)=\left\\{\begin{array}{ll} \frac{c}{x^{2}} & \text { for } x>1 \\ 0 & \text { for } x \leq 1 \end{array}\right. $$ is a density function.

In Problems \(1-4\), determine the sample space for each random experiment. An urn contains six balls numbered \(1-6\), respectively. The random experiment consists of selecting five balls simultaneously without replacement.

Suppose you draw 3 cards from a standard deck of 52 cards. Find the probability that the third card is a club given that the first two cards are clubs.

The following data represent blood cholesterol levels, in \(\mathrm{mg} / \mathrm{dL}\), of patients in a clinical trial: $$ 174,138,212,203,194,245,146,149,164,209,158 $$ Find the median, the sample mean, and the sample variance.

Let \(X\) be a continuous random variable with density \(f(x)\), and assume that \(X \geq 2\). Why is \(E(X) \geq 2 ?\)

An urn contains three green and two blue balls. You remove two balls at random without replacement. Let \(X\) denote the number of green balls in your sample. Find the probability mass function describing the distribution of \(X\).

Let \(X\) be a continuous random variable with density function $$ f(x)=\left\\{\begin{array}{cl} 2 e^{-2 x} & \text { for } x>0 \\ 0 & \text { for } x \leq 0 \end{array}\right. $$ Find \(E(X)\) and \(\operatorname{var}(X)\).

In Problems \(5-8\), assume that $$ \Omega=\\{1,2,3,4,5,6\\} $$ \(A=\\{1,3,5\\}\), and \(B=\\{1,2,3\\}\). Find \(A \cup B\) and \(A \cap B\).

An urn contains five blue and six green balls. You take two balls out of the urn, one after the other, without replacement. Find the probability that the second ball is green given that the first ball is blue.

The Muesli-Mix is a popular breakfast hangout near a campus. A typical breakfast there consists of one beverage, one bowl of cereal, and a piece of fruit. If you can choose among three different beverages, seven different cereals, and four different types of fruit, how many choices for breakfast do you have?

The following data represent the frequency distribution of seed numbers per flower head in a flowering plant: $$ \begin{array}{cc} \hline \text { Seed Number } & \text { Frequency } \\ \hline 9 & 37 \\ 10 & 48 \\ 11 & 53 \\ 12 & 49 \\ 13 & 61 \\ 14 & 42 \\ 15 & 31 \\ \hline \end{array} $$ Calculate the sample mean and the sample variance.

Let \(X\) be uniformly distributed over \((-2,2) .\) Use Chebyshev's inequality to estimate \(P(|X| \geq 1)\), and compare your estimate with the exact answer.

An urn contains five green balls, two blue balls, and three red balls. You remove three balls at random without replacement. Let \(X\) denote the number of red balls. Find the probability mass function describing the distribution of \(X\).

In Problems \(5-8\), assume that $$ \Omega=\\{1,2,3,4,5,6\\} $$ \(A=\\{1,3,5\\}\), and \(B=\\{1,2,3\\}\). Find \(A^{c}\) and show that \(\left(A^{c}\right)^{c}=A\).

An urn contains five green, six blue, and four red balls. You take three balls out of the urn, one after the other, without replacement. Find the probability that the third ball is green given that the first two balls were red.

To study sex differences in food preferences in rats, you offer one of three choices of food to each rat. You plan to have 12 rats for each food-and-sex combination. How many rats will you need?

The following data represent the frequency distribution of the numbers of days that it took a certain ointment to clear up a skin rash: $$ \begin{array}{cc} \hline \text { Number of Days } & \text { Frequency } \\ \hline 1 & 2 \\ 2 & 7 \\ 3 & 9 \\ 4 & 27 \\ 5 & 11 \\ 6 & 5 \\ \hline \end{array} $$ Calculate the sample mean and the sample variance.

You draw 3 cards from a standard deck of 52 cards without replacement. Let \(X\) denote the number of spades in your hand. Find the probability mass function describing the distribution of \(X\).

Let \(X\) be a continuous random variable with distribution function $$ F(x)=\left\\{\begin{array}{cl} 1-\frac{1}{x^{3}} & \text { for } x>1 \\ 0 & \text { for } x \leq 1 \end{array}\right. $$ Find \(E(X)\) and \(\operatorname{var}(X)\).

In Problems \(5-8\), assume that $$ \Omega=\\{1,2,3,4,5,6\\} $$ \(A=\\{1,3,5\\}\), and \(B=\\{1,2,3\\}\). Find \((A \cup B)^{c}\).

A family has two children. The younger one is a girl. Find the probability that the other child is a girl as well.

The genome of the HIV virus consists of 9749 nucleotides. There are four different types of nucleotides. Determine the total number of different genomes of size 9749 nucleotides.

The following data represent the relative frequency distribution of clutch size in a sample of 300 laboratory guinea pigs: $$ \begin{array}{cc} \hline \text { Clutch Size } & \text { Relative Frequency } \\ \hline 2 & 0.05 \\ 3 & 0.09 \\ 4 & 0.12 \\ 5 & 0.19 \\ 6 & 0.23 \\ 7 & 0.12 \\ 8 & 0.13 \\ 9 & 0.07 \\ \hline \end{array} $$ Calculate the sample mean and the sample variance.

Suppose \(X\) is a random variable with mean 10 and variance \(9 .\) What can you say about \(P(|X-10| \geq 5) ?\)

You draw 5 cards from a standard deck of 52 cards without replacement. Let \(X\) denote the number of aces in your hand. Find the probability mass function describing the distribution of \(X\).

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 \(\operatorname{var}(X)\).

In Problems \(5-8\), assume that $$ \Omega=\\{1,2,3,4,5,6\\} $$ \(A=\\{1,3,5\\}\), and \(B=\\{1,2,3\\}\). Are \(A\) and \(B\) disjoint?

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 nucleotides 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 \(\left(5.9736 \times 10^{24} \mathrm{~kg}\right)\) ?

The following data represent the relative frequency distribution of clutch size in a sample of 42 mallards: $$ \begin{array}{cc} \hline \text { Clutch Size } & \text { Relative Frequency } \\ \hline 6 & 0.10 \\ 7 & 0.24 \\ 8 & 0.29 \\ 9 & 0.21 \\ 10 & 0.16 \\ \hline \end{array} $$ Calculate the sample mean and the sample variance.