Probability and Statistics - The Science of Uncertainty, Second Edition CHAPTER 2 Solutions

These are solutions I have come up with for the this textbook, I offer no gaurentee of accuracy

Questions

Let $S=\{1,2,3...\}$ and let $X(s)=s^2$ and $Y(s)=\frac{1}{s}$ for $s\in S$. For each of the following quantities, determine (with explaination) whether or not it exists. If it does exist, then give its value.

- $min_{s\in S}X(s)$
- $max_{s\in S}X(s)$
- $min_{s\in S}Y(s)$
- $max_{s\in S}Y(s)$

Let $S=\{high,middle,low\}$. Define random variables X,Y, and Z by X(high)=-12, X(middle)=-2, X(low)=3, Y(high)=0, Y(middle)=0, Y(low)=1, Z(high)=6, Z(middle)=0, Z(low)=4, Determine whether each of the following relations is true or fasle.

- $X < Y$
- $X\leq Y$
- $Y < Z$
- $Y\leq Z$
- $XY < Z$
- $XY\leq Z$

Let $S=\{1,2,3,4,5\}.$

- Define two different(i.e., nonequal) nonconstant random variablesm X and Y, on S.
- For the random variables X and Y that you have chosen, let $Z=X+Y^2$. Compute Z(s) for all $s\in S$.

Consider rolling a fair six-sided die, so that $S=\{1,2,3,4,5,6\}$. Let $X(s)=s$, and $Y(s)=s^3+2$. Let $Z=XY$. Compute $Z(s)$ for all $s\in S$.

Let A and B be events, and let $X=\textit{I}_A*\textit{I}_B$. Is X an indicator function? If yes, then of what event?

Let $S={\{1,2,3,4\}}$, $X=\textit{I}_{\{1,2\}}$, $Y=\textit{I}_{\{2,3\}}$, $Z=\textit{I}_{\{3,4\}}$. Let $W=X+Y+Z$

- Compute $W(1)$
- Compute $W(2)$
- Compute $W(4)$
- Determine whether or not $W \geq Z$

Let $S={\{1,2,3\}}$, $X=\textit{I}_{\{1\}}$, $Y=\textit{I}_{\{2,3\}}$, $Z=\textit{I}_{\{1,2\}}$. Let $W=X-Y+Z$

- Compute $W(1)$
- Compute $W(2)$
- Compute $W(3)$
- Determine whether or not $W \geq Z$

Let $S={\{1,2,3,4,5\}}$, $X=\textit{I}_{\{1,2,3\}}$, $Y=\textit{I}_{\{2,3\}}$, $Z=\textit{I}_{\{3,4,5\}}$. Let $W=X+Y+Z$

- Compute $W(1)$
- Compute $W(2)$
- Compute $W(5)$
- Determine whether or not $W \geq Z$

Let $S={\{1,2,3,4\}}$, $X=\textit{I}_{\{1,2\}}$, $Y(s)=s^2X(s)$

- Compute $Y(1)$
- Compute $Y(2)$
- Compute $Y(4)$

Let X be a random varriable

- It is necessarily true that $X\geq 0$?
- Is it necessarily true that there is some real number c such that $X+c \geq 0$?
- Suppose the sample space S is finite. Then is it necessarily true that there is some real number c such that $X+c\geq 0$?
- Compute $Y(4)$

Suppose the sample space S is finite. Is it possible to define an unbounded random variiable on S? Why or why not?

Suppose X is a random variable that takes only the vales 0 or 1. Must X be an indicator function? Explain.

Suppose the sample space S is finite, of size m. How many different indicator functions can be defined on S?

Suppose X is a random variable. Let $Y=\sqrt{X}$. Must Y be a radom variable? Explain.

Consider flipping two independent fair coins. Let X be the number of heads that appear. Compute $P(X=x)$ for all real numbers x.

Suppose we flip three fair coins, and let X be the number of heads showing.

- Compute $P(X = x)$ for every real number x.
- Write a formula for $P(X \in \textbf{B})$, for any subset $\textbf{B}$ of the real numbers.

Suppose we roll two fair six-sided dice, and let Y be the sum of the two numbers showing.

- Compute $P(Y = y)$ for every real number y.
- Write a formula for$ P(Y \in\textbf{B} )$, for any subset $\textbf{B}$ of the real numbers.

Suppose we roll one fair six-sided die, and let Z be the number showing. Let $W = Z^3 + 4$, and let $V = \sqrt{Z}$.

- Compute $P(W = w)$ for every real number w
- Compute $P(V = v)$ for every real number v.
- Compute $P(ZW = x)$ for every real number x.
- Compute $P(VW = y)$ for every real number y.
- Compute $P(V + W = r )$ for every real number r.

Suppose that a bowl contains 100 chips: 30 are labelled 1, 20 are labelled 2, and 50 are labelled 3. The chips are thoroughly mixed, a chip is drawn, and the number X on the chip is noted.

- Compute $P(X = x)$ for every real number x.
- Suppose the first chip is replaced, a second chip is drawn, and the number Y on the chip noted. Compute $P(Y = y)$ for every real number y.
- Compute $P(W = w$) for every real number w when $W = X + Y$.

Suppose a standard deck of 52 playing cards is thoroughly shuffled and a single card is drawn. Suppose an ace has value 1, a jack has value 11, a queen has value 12, and a king has value 13.

- Compute $P(X = x)$ for every real number x, when X is the value of the card drawn.
- Suppose that Y = 1, 2, 3, or 4 when a diamond, heart, club, or spade is drawn. Compute P(Y = y) for every real number y.
- Compute $P(W = w)$ for every real number w when $W = X + Y$.

Suppose a university is composed of 55% female students and 45% male students. A student is selected to complete a questionnaire. There are 25 questions on the questionnaire administered to a male student and 30 questions on the questionnaire administered to a female student. If X denotes the number of questions answered by a randomly selected student, then compute $P(X = x)$ for every real number x.

Suppose that a bowl contains 10 chips, each uniquely numbered 0 through 9. The chips are thoroughly mixed, one is drawn and the number on it, $X_1$, is noted. This chip is then replaced in the bowl. A second chip is drawn and the number on it, $X_2$, is noted. Compute $P(W = w)$ for every real number w when $W = X_1 + 10X_2$.

Consider rolling two fair six-sided dice. Let Y be the sum of the numbers showing. What is the probability function of Y?

Consider flipping a fair coin. Let Z = 1 if the coin is heads, and Z = 3 if the coin is tails. Let $W = Z^2 + Z$.

- What is the probability function of Z?
- What is the probability function of W?

Consider flipping two fair coins. Let X = 1 if the first coin is heads, and X = 0 if the first coin is tails. Let Y = 1 if the second coin is heads, and Y = 5 if the second coin is tails. Let $Z = XY$. What is the probability function of Z?

Consider flipping two fair coins. Let $X = 1$ if the first coin is heads, and $X = 0$ if the first coin is tails. Let $Y = 1$ if the two coins show the same thing (i.e., both heads or both tails), with $Y = 0$ otherwise. Let $Z = X + Y$, and $W = XY$.

- What is the probability function of Z?
- What is the probability function of W?

Consider rolling two fair six-sided dice. Let W be the product of the numbers showing. What is the probability function of W?

Let $Z$ ~ $Geometric(\theta )$. Compute $P(5 \leq Z \leq 9)$.

Let $X$ ~ $Binomial(12, \theta )$. For what value of $\theta$ is $P(X = 11)$ maximized?

Let $W$ ~ $Poisson(\lambda )$. For what value of $\lambda$ is $P(W = 11)$ maximized?

Let $Z$ ~ $Negative-Binomial(3, 1/4)$. Compute $P(Z \leq 2)$.

Let $X$ ~ $Geometric(1/5)$. Compute $P(X^2 \leq 15)$.

Let $Y$ ~ $Binomial(10,\theta)$. Compute $P(Y = 10)$.

Let $X$ ~ $Poisson(\lambda)$. Let $Y = X - 7$. What is the probability function of Y?

Let $X$ ~ $Hypergeometric(20, 7, 8)$. What is the probability that $X = 3$? What is the probability that $X = 8$?

Suppose that a symmetrical die is rolled 20 independent times, and each time we record whether or not the event $\{2, 3, 5, 6\}$ has occurred

- What is the distribution of the number of times this event occurs in 20 rolls?
- Calculate the probability that the event occurs five times.

Suppose that a basketball player sinks a basket from a certain position on the court with probability 0.35.

- What is the probability that the player sinks three baskets in 10 independent throws?
- What is the probability that the player throws 10 times before obtaining the first basket?
- What is the probability that the player throws 10 times before obtaining two baskets?

An urn contains 4 black balls and 5 white balls. After a thorough mixing, a ball is drawn from the urn, its color is noted, and the ball is returned to the urn.

- What is the probability that 5 black balls are observed in 15 such draws?
- What is the probability that 15 draws are required until the first black ball is observed?
- What is the probability that 15 draws are required until the fifth black ball is observed?

An urn contains 4 black balls and 5 white balls. After a thorough mixing, a ball is drawn from the urn, its color is noted, and the ball is set aside. The remaining balls are then mixed and a second ball is drawn.

- What is the probability distribution of the number of black balls observed?
- What is the probability distribution of the number of white balls observed?

(Poisson processes and queues) Consider a situation involving a server, e.g., a cashier at a fast-food restaurant, an automatic bank teller machine, a telephone exchange, etc. Units typically arrive for service in a random fashion and form a queue when the server is busy. It is often the case that the number of arrivals at the server, for some specific unit of time t, can be modeled by a $Poisson(\lambda t)$ distribution and is such that the number of arrivals in nonoverlapping periods are independent. In Chapter 3, we will show that $\lambda t$ is the average number of arrivals during a time period of length $t$, and so $\lambda$ is the rate of arrivals per unit of time. Suppose telephone calls arrive at a help line at the rate of two per minute. A Poisson process provides a good model.

- What is the probability that five calls arrive in the next 2 minutes?
- What is the probability that five calls arrive in the next 2 minutes and then five more calls arrive in the following 2 minutes?
- What is the probability that no calls will arrive during a 10-minute period?

Suppose an urn contains 1000 balls - one of these is black, and the other 999 are white. Suppose that 100 balls are randomly drawn from the urn with replacement. Use the appropriate Poisson distribution to approximate the probability that five black balls are observed.

Suppose that there is a loop in a computer program and that the test to exitthe loop depends on the value of a random variable X. The program exits the loop whenever $X \in \mathbb{A}$, and this occurs with probability $1/3$. If the loop is executed at least once, what is the probability that the loop is executed five times before exiting?

U ~ $Uniform[0, 1]$. Compute each of the following.

- $P(U\leq 0)$
- $P(U=\frac{1}{2})$
- $P(U < -\frac{1}{3})$
- $P(U\leq \frac{2}{3})$
- $P(U < \frac{2}{3})$
- $P(U < 1)$
- $P(U\leq 17)$

U ~ $Uniform[1, 4]$. Compute each of the following.

- $P(W \geq 5)$
- $P(W \geq 2)$
- $P(W^2 \geq 9)$ (Hint: If $W^2 \leq 9$, what must W be?)
- $P(W^2 \leq 2)$

Let Z ~ $Exponential(4)$. Compute each of the following.

- $P(Z \geq 5)$
- $P(Z \geq -5)$
- $P(Z^2 \geq 9)$
- $P(Z^4-17 \geq 9)$

Establish for which constants c the following functions are densities.

- $f(x) = cx$ on $(0, 1)$ and 0 otherwise.
- $f(x) = cx^n$ on $(0, 1)$ and 0 otherwise, for n a nonnegative integer.
- $f(x) = cx^{\frac{1}{2}}$ on $(0, 2)$ and 0 otherwise.
- $f(x) =c*sin(x)$ on $(0, \frac{\pi}{2})$ and 0 otherwise.

Is the function defined by $f (x) =\frac{x}{3}$ for $-1 < x < 2$ and 0 otherwise, a density? Why or why not?

Let X ~ $Exponential(3)$. Compute each of the following.

- $P(0 < X < 1)$
- $P(0 < X < 3)$
- $P(0 < X < 5)$
- $P(2 < X < 5)$
- $P(2 < X < 10)$
- $P(X > 2)$