Question 1

Suppose $X$ is a discrete random variable with probability mass function $p(x)$. Which of the following correctly expresses the expected value $E[X]$?

Accepted Answer

$E[X] = \sum_{x} x p(x)$

Question 2

A random variable $X$ has the probability mass function $p(i) = \binom{n}{i} p^i (1-p)^{n-i}$ for $i = 0, 1, ..., n$. What is the variance of $X$?

Accepted Answer

$Var(X) = np(1-p)$

Question 3

Suppose $X$ is a Poisson random variable with parameter $\lambda$. Which of the following statements is true?

Accepted Answer

$E[X] = Var(X) = \lambda$

Question 4

A fair coin is tossed 10 times. What is the probability that exactly 6 heads occur?

Accepted Answer

$\binom{10}{6} \left(\frac{1}{2}\right)^{10}$

Question 5

Let $X$ be a geometric random variable with parameter $p$. What is the expected value $E[X]$?

Accepted Answer

$E[X] = \frac{1}{p}$

Question 6

A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same color, you win $1.10; if they are different colors, you lose $1.00. What is the expected value of your winnings?

Accepted Answer

$E[X] = -0.02$

Question 7

Suppose $X$ is a continuous random variable with probability density function $f(x) = 2x$ for $0 \leq x \leq 1$, and $f(x) = 0$ otherwise. What is $E[X]$?

Accepted Answer

$E[X] = \frac{2}{3}$

Question 8

A random variable $X$ is uniformly distributed on $(0, 10)$. What is $Var(X)$?

Accepted Answer

$Var(X) = \frac{100}{12}$

Question 9

Suppose the lifetime (in hours) of a certain device is exponentially distributed with mean 100. What is the probability that the device lasts more than 200 hours?

Accepted Answer

$e^{-2}$

Question 10

Let $X$ be a normal random variable with mean $\mu = 3$ and variance $\sigma^2 = 9$. What is $P\{X > 0\}$?

Accepted Answer

$\Phi(1)$

Question 11

A random variable $X$ has the probability density function $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$. Which of the following is true about the exponential distribution?

Accepted Answer

It is memoryless: $P\{X > s + t | X > t\} = P\{X > s\}$ for all $s, t \geq 0$

Question 12

Suppose $X$ is a binomial random variable with parameters $n = 100$ and $p = 0.5$. Using the normal approximation, what is the probability that $X \geq 60$? (Use the continuity correction.)

Accepted Answer

$P\left(Z \geq \frac{59.5 - 50}{5}\right)$

Question 13

A hypergeometric random variable $X$ with parameters $n, N, m$ represents the number of white balls selected when $n$ balls are randomly chosen from an urn containing $N$ balls, $m$ of which are white. What is the expected value $E[X]$?

Accepted Answer

$E[X] = n \frac{m}{N}$

Question 14

Suppose $X$ is a continuous random variable with density $f(x)$ and $Y = aX + b$ for constants $a$ and $b$. Which of the following is true?

Accepted Answer

$E[Y] = aE[X] + b$

Question 15

A random variable $X$ is said to have a Laplace distribution with parameter $\lambda > 0$ if its density is $f(x) = \frac{1}{2\lambda} e^{-\lambda |x|}$. Which of the following statements is true?

Accepted Answer

The Laplace distribution is symmetric about zero.

Question 16

Suppose $X$ is a gamma random variable with parameters $\alpha$ and $\lambda$. What is $E[X]$?

Accepted Answer

$E[X] = \frac{\alpha}{\lambda}$

Discrete and Continuous Random Variables: Probability Distributions, Expectation, and Applications

Study Guide - Practice Questions

Study Guide - Flashcards