Question 1

Which of the following best defines a discrete random variable?

Accepted Answer

A variable that maps the sample space of a probability experiment to a subset of natural numbers or integers.

Question 2

Suppose you toss a fair coin three times. Let $$W be $$the random variable representing the number of tails minus the number of heads. What is a possible value of $$W$$?

Accepted Answer

$$W = 0$$

Question 3

Which of the following is NOT a property of the probability mass function (pmf) $$p(x_i) = P(X = x_i)$$ for a discrete random variable $$X$$?

Accepted Answer

$$P(X = x_i)$$ can be negative for some $$i$$

Question 4

A shipment of 7 television sets contains 2 defective sets. A hotel randomly purchases 4 sets. Let $$X be $$the number of defective sets purchased. Which probability distribution is appropriate for $$X$$?

Accepted Answer

Hypergeometric distribution

Question 5

A car agency sells 50% of its inventory of a certain foreign car equipped with side airbags. What is the probability that exactly 2 out of the next 4 cars sold have side airbags?

Accepted Answer

$$P(X = 2) = \binom{4}{2} (0.5)^2$ (0.5)^2$

Question 6

Which formula represents the probability mass function for a Poisson random variable with parameter $$\lambda$$?

Accepted Answer

$$P(X = k) = e$$^{-\lambda}$$ \lambda^k / k!$$

Question 7

Let $$X be a $$discrete random variable with $$P(X = 1) = 0.4$$, $$P(X = 2) = 0.2$$, $$P(X = 3) = 0.3$$, $$P(X = 4) = 0.1. $$What is $$F(3)$$, the cumulative distribution function at $$x = 3$$?

Accepted Answer

$$F(3) = 0.9$$

Question 8

Which of the following is NOT a property of the cumulative distribution function (CDF) $$F(x)$$ for a discrete random variable?

Accepted Answer

$$F(x)$$ can decrease for some values of $$x$$

Question 9

You roll two fair 6-sided dice and let $$X be $$the sum of the two dice. What is $$F(4)$$, the cumulative probability that $$X \leq 4$$?

Accepted Answer

$$F(4) = \frac{1}{36} + \frac{2}{36} + \frac{3}{36} = \frac{6}{36}$$

Question 10

Given the probability distribution for imperfections per 10 meters of fabric: $$P(X=0)=0.38$$, $$P(X=1)=0.36$$, $$P(X=2)=0.16$$, $$P(X=3)=0.07$$, $$P(X=4)=0.03$$, what is $$F(2)$$?

Accepted Answer

$$F(2) = 0.38 + 0.36 + 0.16 = 0.90$$

Question 11

Which of the following is the correct formula for the survival function $$S(x)$$ for a discrete random variable $$X$$?

Accepted Answer

$$S(x) = 1 - F(x)$$

Question 12

Which of the following scenarios is best modeled by a geometric distribution?

Accepted Answer

The number of trials until the first success in a sequence of independent Bernoulli trials.

Question 13

If $$X is a $$binomial random variable with parameters $$n = 4$$ and $$p = 0.5$$, what is $$P(X = 3)$$?

Accepted Answer

$$P(X = 3) = \binom{4}{3} (0.5)^3$ (0.5)^1$

Question 14

Which of the following is a property of the probability mass function (pmf) for a discrete random variable?

Accepted Answer

$$\sum_i P(X = x_i) = 1$$

Question 15

A hotel purchases 4 TVs from a shipment of 7, 2 of which are defective. What is the probability that exactly 1 defective TV is purchased? (Use the hypergeometric formula.)

Accepted Answer

$$P(X = 1) = \frac{\binom{2}{1} \binom{5}{3}}{\binom{7}{4}}$$

Question 16

Which of the following is the correct formula for the negative binomial probability mass function?

Accepted Answer

$$P(X = k) = \binom{n+k-1}{k} p^n (1-p)^k$$

Discrete Random Variables and Probability Distributions

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