Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable that has a finite or countable number of possible outcomes.

Question 2

Let $$x$$ represent the number of defective light bulbs in a box of 100. Is $$x$$ discrete or continuous?

Accepted Answer

Discrete, because the number of defective bulbs can be counted.

Question 3

Which of the following is NOT a requirement for a discrete probability distribution?

Accepted Answer

Each outcome must be equally likely.

Question 4

Given the following probabilities for a random variable $$x$$: $$P(x=1)=0.3$$, $$P(x=2)=0.5$$, $$P(x=3)=0.2. Is $$this a valid probability distribution?

Accepted Answer

Yes, because all probabilities are between 0 and 1 and their sum is 1.

Question 5

Suppose a probability distribution has $$P(x=1)=0.4$$, $$P(x=2)=0.7$$, $$P(x=3)=-0.1. $$Why is this NOT a valid probability distribution?

Accepted Answer

Because the probabilities are not all between 0 and 1.

Question 6

How do you calculate the mean $$\mu of a $$discrete probability distribution?

Accepted Answer

$$\mu = \sum x \cdot P(x)$$

Question 7

A company tests 200 batteries and finds the following probability distribution for the number of batteries that last more than 10 hours: $$P(x=0)=0.1$$, $$P(x=1)=0.3$$, $$P(x=2)=0.6. $$What is the expected number of batteries that last more than 10 hours?

Accepted Answer

$$0 \cdot 0.1 + 1 \cdot 0.3 + 2 \cdot 0.6 = 1.5$$

Question 8

Which formula correctly represents the variance $$\$$sigma^2$$ of a discrete probability distribution?

Accepted Answer

$$\$$sigma^2$$ = \sum (x - \mu)^2$ \cdot P(x)$$

Question 9

A raffle sells 1000 tickets at $$3 each for a single prize of 2000. $$What is the expected value of buying one ticket?

Accepted Answer

$$E(x) = (2000-3) \cdot \frac{1}{1000} + (-3) \cdot \frac{999}{1000} = -1.00$$

Question 10

Which of the following best describes the expected value of a discrete random variable?

Accepted Answer

The long-run average value of repetitions of the experiment.

Question 11

A probability distribution is given by $$P(x=1)=0.2$$, $$P(x=2)=0.5$$, $$P(x=3)=0.3. $$What is the standard deviation if the mean is $$\mu=2.1$$?

Accepted Answer

$$\sqrt{(1-2.1)^2$ \cdot 0.2 + (2-2.1)^2$ \cdot 0.5 + (3-2.1)^2$ \cdot 0.3} = 0.7$$

Question 12

Which of the following is a real-life example of a continuous random variable?

Accepted Answer

The amount of milk in a 1-liter carton.

Question 13

If a probability distribution has $$P(x=1)=0.4$$, $$P(x=2)=0.3$$, $$P(x=3)=0.2$$, $$P(x=4)=0.1$$, what is the mean?

Accepted Answer

$$1 \cdot 0.4 + 2 \cdot 0.3 + 3 \cdot 0.2 + 4 \cdot 0.1 = 2.0$$

Question 14

A student claims that a probability distribution is valid because all probabilities are positive. What is missing from their reasoning?

Accepted Answer

They forgot to check if the sum of the probabilities is 1.

Question 15

Which of the following statements about the variance of a discrete probability distribution is true?

Accepted Answer

Variance measures the spread of the distribution around the mean.

Question 16

A game costs $$5 to play. You have a 1 in 100 chance to win 400. $$What is your expected gain or loss?

Accepted Answer

$$E(x) = (400-5) \cdot \frac{1}{100} + (-5) \cdot \frac{99}{100} = -1.00$$

Question 17

Which of the following is NOT a step in constructing a discrete probability distribution from frequency data?

Accepted Answer

Multiply each frequency by the mean.

Question 18

If the standard deviation of a discrete probability distribution is small, what does this indicate about the data?

Accepted Answer

The data values are clustered closely around the mean.

Discrete Probability Distributions: Study Notes for Introductory Statistics

Study Guide - Practice Questions

Study Guide - Flashcards