Q: Given a discrete probability distribution with $$x$$ values 0, 1, 2 and corresponding probabilities 0.2, 0.5, and 0.3, what is the mean $$\mu of $$the distribution?

$$\mu = 0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 1.1$$

Q: In a binomial experiment with $$n = 5$$ trials and probability of success $$p = 0.4$$, what is the probability of exactly 3 successes? Use the binomial formula $$P(x) = nCx p^x q$$^{n-x}$$ where q = 1 - p$.

$$P(3) = 5C3 \times 0.4^3$ \times 0.6^2$ = 10 \times 0.064 \times 0.36 = 0.2304$$

Q: Suppose the probability of a business failing in a year is $$p = 0.2. $$What is the probability that the first failure occurs on the third business selected? Use the geometric formula $$P(x) = p q$$^{x-1}$$.

$$P(3) = 0.2 \times 0.8$$^{2}$$ = 0.128$$

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 that can be listed.

Question 2

Let $$x$$ represent the number of students who pass a statistics exam in a class of 30. Is $$x a $$discrete or continuous random variable?

Accepted Answer

Discrete, because the number of students who pass can be counted.

Question 3

Which of the following must be true for a discrete probability distribution?

Accepted Answer

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

Question 4

Given a discrete probability distribution with $$x$$ values 0, 1, 2 and corresponding probabilities 0.2, 0.5, and 0.3, what is the mean $$\mu of $$the distribution?

Accepted Answer

$$\mu = 0 	imes 0.2 + 1 	imes 0.5 + 2 	imes 0.3 = 1.1$$

Question 5

A binomial experiment is characterized by which of the following?

Accepted Answer

A fixed number of independent trials, each with two possible outcomes and constant probability of success.

Question 6

In a binomial experiment with $$n = 5$$ trials and probability of success $$p = 0.4$$, what is the probability of exactly 3 successes? Use the binomial formula $$P(x) = nCx p^x q$$^{n-x}$$ where q = 1 - p$.

Accepted Answer

$$P(3) = 5C3 	imes 0.4^3$ 	imes 0.6^2$ = 10 	imes 0.064 	imes 0.36 = 0.2304$$

Question 7

If a binomial experiment has $$n = 10$$ trials and $$p = 0.3$$, what is the mean $$\mu$$ and variance $$\$$sigma^2$$ of the distribution?

Accepted Answer

$$\mu = 3, \$$sigma^2$$ = 2.1$$

Question 8

A geometric distribution models the probability that the first success occurs on which trial?

Accepted Answer

On trial $$x$$, where $$P(x) = p q$$^{x-1}$$ and q = 1 - p$.

Question 9

Suppose the probability of a business failing in a year is $$p = 0.2. $$What is the probability that the first failure occurs on the third business selected? Use the geometric formula $$P(x) = p q$$^{x-1}$$.

Accepted Answer

$$P(3) = 0.2 	imes 0.8$$^{2}$$ = 0.128$$

Question 10

Which scenario is best modeled by a Poisson distribution?

Accepted Answer

Counting the number of emails received in an hour.

Question 11

If the mean number of phone calls received per hour at a call center is $$\mu = 5$$, what is the probability that exactly 2 calls are received in an hour? Use the Poisson formula $$P(x) = \frac{\mu^x e$$^{-\mu}$$}{x!}$$ with $$e = 2.71818.$$

Accepted Answer

$$P(2) = \frac{5^2$ 	imes e$$^{-5}$$}{2!} = \frac{25 	imes 0.00674}{2} = 0.084$$

Question 12

A discrete probability distribution has $$x$$ values 1, 2, 3 with probabilities 0.2, 0.5, and 0.3. What is the variance $$\$$sigma^2$$? Use $$\$$sigma^2$$ = \sum (x - \$$mu)^2 P(x$$)$$.

Accepted Answer

$$\$$sigma^2$$ = (1-1.1)^2$ 	imes 0.2 + (2-1.1)^2$ 	imes 0.5 + (3-1.1)^2$ 	imes 0.3 = 0.02 + 0.405 + 1.083 = 1.508$$

Question 13

A survey finds that 30% of people prefer tea over coffee. If you randomly select 6 people, what is the probability that exactly 2 prefer tea? Use $$P(x) = nCx p^x q$$^{n-x}$$ with p = 0.3$, q = 0.7$, n = 6$, x = 2$.

Accepted Answer

$$P(2) = 6C2 	imes 0.3^2$ 	imes 0.7^4$ = 15 	imes 0.09 	imes 0.2401 = 0.324$$

Question 14

Which of the following statements about the expected value $$E(x) of a $$discrete random variable is correct?

Accepted Answer

It is calculated as $$E(x) = \sum x P(x).$$

Question 15

A Poisson distribution has a mean $$\mu = 4. $$What is the probability of exactly 0 occurrences in an interval? Use $$P(0) = \frac{\mu^0$ e$$^{-\mu}$$}{0!}.$$

Accepted Answer

$$P(0) = 1 	imes e$$^{-4}$$ = 0.018$$

Question 16

Which of the following is NOT a requirement for a binomial experiment?

Accepted Answer

The probability of success must change from trial to trial.

Question 17

A hospital records the number of patients admitted each day. If the average is $$\mu = 8$$ patients per day, what is the probability that exactly 10 patients are admitted on a given day? Use $$P(x) = \frac{\mu^x e$$^{-\mu}$$}{x!}.$$

Accepted Answer

$$P(10) = \frac{8$$^{10}$$ e$$^{-8}$$}{10!} = 0.099$$

Question 18

If a binomial experiment has $$n = 12$$ trials and $$p = 0.5$$, what is the standard deviation $$\sigma$$? Use $$\sigma = \sqrt{npq}.$$

Accepted Answer

$$\sigma = \sqrt{12 	imes 0.5 	imes 0.5} = \sqrt{3} = 1.732$$

Discrete Probability Distributions: Random Variables, Binomial, Geometric, and Poisson Distributions

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