$$E(X) = \frac{500 - 2}{1000} + \frac{-2 \times 999}{1000}$$

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 products found in a sample of 10. Is $$X$$ discrete or continuous, and why?

Accepted Answer

Discrete, because the number of defective products can only be whole numbers.

Question 3

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

Accepted Answer

Each probability is greater than 1.

Question 4

Given the following probability distribution, is it valid?

$$X: 1, 2, 3, 4$$ 
$$P(X): 0.2, 0.3, 0.4, 0.2$$

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

A probability distribution for $$X is $$given as follows: $$P(1) = 0.1$$, $$P(2) = 0.3$$, $$P(3) = 0.6. $$What is the mean $$\mu$$?

Accepted Answer

$$\mu = 2.4$$

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$ P(x)$$

Question 8

Suppose the mean score on a test is $$\mu = 3$$ and the variance is $$\$$sigma^2$$ = 1.5. $$What is the standard deviation?

Accepted Answer

$$\sigma = \sqrt{1.5}$$

Question 9

A company sells 1000 raffle tickets at $$2 each. There is one prize of 500. $$What is the expected value of buying one ticket?

Accepted Answer

$$E(X) = \frac{500 - 2}{1000} + \frac{-2 	imes 999}{1000}$$

Question 10

Which of the following scenarios describes a continuous random variable?

Accepted Answer

The amount of milk in a 1-gallon jug.

Question 11

If a probability distribution has $$P(1) = 0.3$$, $$P(2) = 0.5$$, $$P(3) = 0.1$$, and $$P(4) = 0.1$$, is this a valid probability distribution?

Accepted Answer

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

Question 12

A discrete probability distribution is given by $$P(1) = 0.2$$, $$P(2) = 0.5$$, $$P(3) = 0.3. $$What is the expected value?

Accepted Answer

$$E(X) = 2.1$$

Question 13

Which of the following best interprets the expected value in a lottery scenario?

Accepted Answer

The average amount you can expect to win per ticket in the long run.

Question 14

If $$X is a $$discrete random variable with $$P(X=1) = 0.4$$, $$P(X=2) = 0.4$$, $$P(X=3) = 0.2$$, what is $$P(X \geq 2)$$?

Accepted Answer

$$0.6$$

Question 15

A probability distribution is given as $$P(1) = 0.3$$, $$P(2) = 0.3$$, $$P(3) = 0.3$$, $$P(4) = 0.1. $$What is the variance if the mean is $$2.2$$?

Accepted Answer

$$\$$sigma^2$$ = \sum (x - 2.2)^2$ P(x)$$

Question 16

Which of the following is a step in constructing a discrete probability distribution from frequency data?

Accepted Answer

Divide the frequency of each outcome by the total number of observations.

Discrete Probability Distributions: Random Variables, Mean, Variance, and Expected Value

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