Q: A probability distribution is given as $$P(X=1)=0.2,\ P(X=2)=0.5,\ P(X=3)=0.3. $$What is the mean $$\mu$$?

$$\mu = 1 \times 0.2 + 2 \times 0.5 + 3 \times 0.3 = 2.1$$

Q: If a discrete random variable $$X$$ has the probability distribution $$P(X=0)=0.1,\ P(X=1)=0.6,\ P(X=2)=0.3$$, what is the variance $$\$$sigma^2$$?

$$\$$sigma^2$$ = (0-1.2)^2$ \times 0.1 + (1-1.2)^2$ \times 0.6 + (2-1.2)^2$ \times 0.3 = 0.36$$

Question 1

Which of the following is a requirement for a discrete probability distribution?

Accepted Answer

Each probability must be between 0 and 1, inclusive, and the sum of all probabilities must equal 1.

Question 2

A random variable $$X$$ represents the number of heads when tossing two coins. What is the probability distribution for $$X$$?

Accepted Answer

$$P(X=0)=\frac{1}{4},\ P(X=1)=\frac{1}{2},\ P(X=2)=\frac{1}{4}$$

Question 3

Which of the following is an example of a continuous random variable?

Accepted Answer

The height of a randomly selected student.

Question 4

Given the following probability distribution, is it valid? $$P(X=1)=0.3,\ P(X=2)=0.5,\ P(X=3)=0.4$$

Accepted Answer

No, because the sum of the probabilities is greater than 1.

Question 5

How do you calculate the mean (expected value) $$\mu of a $$discrete probability distribution?

Accepted Answer

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

Question 6

A company gives a personality test to 150 employees, scoring from 1 (passive) to 5 (aggressive). If 30 employees score 1, what is the probability that a randomly selected employee scores 1?

Accepted Answer

$$P(X=1) = \frac{30}{150}$$

Question 7

Which formula is used to calculate the variance $$\$$sigma^2$$ of a discrete probability distribution?

Accepted Answer

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

Question 8

Suppose you buy a $$2$$ raffle ticket for a drawing with 1500 tickets and prizes of $$500, 250$$, $$150, and 75. $$What is the expected value of your winnings?

Accepted Answer

$$E(X) = \frac{500+250+150+75}{1500} - 2$$

Question 9

Which of the following statements is TRUE about the standard deviation $$\sigma of a $$discrete probability distribution?

Accepted Answer

$$\sigma is $$the square root of the variance.

Question 10

A probability distribution is given as $$P(X=1)=0.2,\ P(X=2)=0.5,\ P(X=3)=0.3. $$What is the mean $$\mu$$?

Accepted Answer

$$\mu = 1 	imes 0.2 + 2 	imes 0.5 + 3 	imes 0.3 = 2.1$$

Question 11

Which of the following is NOT a property of a probability distribution?

Accepted Answer

Probabilities can be negative as long as their sum is 1.

Question 12

If a discrete random variable $$X$$ has the probability distribution $$P(X=0)=0.1,\ P(X=1)=0.6,\ P(X=2)=0.3$$, what is the variance $$\$$sigma^2$$?

Accepted Answer

$$\$$sigma^2$$ = (0-1.2)^2$ 	imes 0.1 + (1-1.2)^2$ 	imes 0.6 + (2-1.2)^2$ 	imes 0.3 = 0.36$$

Question 13

Which scenario best illustrates the concept of expected value in a real-world context?

Accepted Answer

Calculating the average amount you expect to win or lose per ticket in a lottery over many drawings.

Question 14

A distribution lists $$P(X=1)=0.4,\ P(X=2)=0.3,\ P(X=3)=0.2,\ P(X=4)=0.1. Is $$this a probability distribution?

Accepted Answer

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

Question 15

Which of the following best describes a discrete random variable?

Accepted Answer

It can only take on specific, countable values.

Question 16

If the mean of a discrete probability distribution is $$\mu = 2.5$$ and the variance is $$\$$sigma^2$$ = 1.25$$, what is the standard deviation?

Accepted Answer

$$\sigma = \sqrt{1.25} \approx 1.12$$

Question 17

A probability distribution for $$X is$$ $$P(X=1)=0.5,\ P(X=2)=0.3,\ P(X=3)=0.2. $$What is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 1 	imes 0.5 + 2 	imes 0.3 + 3 	imes 0.2 = 1.7$$

Question 18

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.

Discrete Probability Distributions: Study Guide for Elementary Statistics

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