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 Two marbles are withdrawn randomly.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 \u003e 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: $$P\{X \u003e s + t | X \u003e t\} = P\{X \u003e s\}$$ 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 $$\lambda \u003e 0 if $$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

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