Chapter 12

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 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 \(X\).

Determine the sample space for each random experiment. The random experiment consisting of tossing a coin three times.

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

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 levels and two light levels. For each combination of fertilizer level and light level, you want four 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.

Let \(X\) be uniformly distributed over \((1,4)\). (a) Use Markov's inequality to estimate \(P(X \geq a), 1 \leq a \leq 4\), and compare your estimate with the exact answer. (b) Find the value of \(a \in(1,4)\) that minimizes the difference between the bound and the exact probability computed in (a).

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 \(X\).

Determine the sample space for each random experiment. The random experiment consisting of rolling a six-sided die twice.

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

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 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.

Prove Markov's inequality when \(X\) is a nonnegative discrete random variable with \(E(X)<\infty\)

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

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.

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

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 beetle, is an important predator of egg masses of Ostrinia nubilalis, the European corn borer. \(C\). maculata also feeds on aphids and maize pollen. To study its food preferences, you choose two ages of \(C\). maculata beetles and all 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 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.

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 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.

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.

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 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 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 \(X\).

Assume that $$ \Omega=[1,2,3,4,5,6\\} $$ \(A=\\{1,3,5\\}\), and \(B=\\{1,2,3\\}\) Find \(A \cup B\) and \(A \cap B\).

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

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 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.

Let \(X\) be standard normally distributed. Use Chebyshev's inequality to estimate (a) \(P(|X| \geq 1)\), (b) \(P(|X| \geq 2)\), and (c) \(P(|X| \geq 3)\). Compare each 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 \(X\).

Assume that $$ \Omega=[1,2,3,4,5,6\\} $$ \(A=\\{1,3,5\\}\), and \(B=\\{1,2,3\\}\) Find \(A^{c}\) and show that \(\left(A^{c}\right)^{c}=A\).

Let \(X\) be a continuous random variable with density function \(f(x)=\frac{1}{2} e^{-|x|}\) for \(x \in \mathbf{R}\). Find \(E(X)\) and \(\operatorname{var}(X)\).

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 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 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 \(X\).

Assume that $$ \Omega=[1,2,3,4,5,6\\} $$ \(A=\\{1,3,5\\}\), and \(B=\\{1,2,3\\}\) Find \((A \cup B)^{c}\).

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

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 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.

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 ?

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 \(X\).