Q: A student guesses on all 10 questions of a true/false quiz. What is the probability the student gets exactly 8 correct? (Use $$n=10$$, $$p=0.5)$$

$$P(X=8) = \left( \begin{array}{c} 10 \\ 8 \end{array} \right) (0.5)^8$ (0.5)^2$

Q: Suppose the probability that a randomly selected household is wireless-only is $$p=0.25. In a $$sample of $$n=20$$ households, what is the probability that exactly 5 are wireless-only?

$$P(X=5) = \left( \begin{array}{c} 20 \\ 5 \end{array} \right) (0.25)^5$ (0.75$$)^{15}$$

Q: If $$X is a $$binomial random variable with $$n=400$$ and $$p=0.35$$, what is the mean and standard deviation?

$$\mu_X = 140$$, $$\sigma_X = \sqrt{400 \times 0.35 \times 0.65}$$

Question 1

Which of the following is a discrete random variable?

Accepted Answer

The number of heads in five coin flips.

Question 2

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

Accepted Answer

The probability of success changes from trial to trial.

Question 3

Given the probability distribution below, what is the mean (expected value) of $$X$$?

$$x$$: 0, 1, 2, 3
$$P(x)$$: 0.1, 0.4, 0.3, 0.2

Accepted Answer

$$\mu_X = 1.6$$

Question 4

A student guesses on all 10 questions of a true/false quiz. What is the probability the student gets exactly 8 correct? (Use $$n=10$$, $$p=0.5)$$

Accepted Answer

$$P(X=8) = \left( \begin{array}{c} 10 \ 8 \end{array} ight) (0.5)^8$ (0.5)^2$

Question 5

Which formula gives the standard deviation of a discrete random variable $$X$$ with mean $$\mu_X$$?

Accepted Answer

$$\sigma_X = \sqrt{\sum (x - \mu_X)^2$ P(x)}$$

Question 6

A Poisson process has an average rate of $$\lambda = 4$$ per hour. What is the probability of exactly 2 occurrences in one hour?

Accepted Answer

$$P(X=2) = \frac{4^2$}{2!} e$$^{-4}$$

Question 7

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

Accepted Answer

The number of emails received per hour.

Question 8

Suppose the probability that a randomly selected household is wireless-only is $$p=0.25. In a $$sample of $$n=20$$ households, what is the probability that exactly 5 are wireless-only?

Accepted Answer

$$P(X=5) = \left( \begin{array}{c} 20 \ 5 \end{array} ight) (0.25)^5$ (0.75$$)^{15}$$

Question 9

If $$X is a $$binomial random variable with $$n=400$$ and $$p=0.35$$, what is the mean and standard deviation?

Accepted Answer

$$\mu_X = 140$$, $$\sigma_X = \sqrt{400 	imes 0.35 	imes 0.65}$$

Question 10

A chocolate bar has 0.6 insect fragments per gram. What is the mean number of fragments in a 10-gram sample?

Accepted Answer

$$\mu_X = 0.6 	imes 10 = 6$$

Question 11

Which of the following is a property of a probability distribution for a discrete random variable $$X$$?

Accepted Answer

$$\sum P(x) = 1$$

Question 12

A biologist finds that 2000 beetles are evenly spread over 1000 square feet, divided into 200 equal subsections. What is the expected number of beetles per subsection?

Accepted Answer

$$\frac{2000}{200} = 10$$

Question 13

Which of the following best describes the Law of Large Numbers as it relates to expected value?

Accepted Answer

As the number of trials increases, the sample mean approaches the expected value.

Question 14

A Poisson random variable $$X$$ has mean $$\lambda t. $$What is the standard deviation of $$X$$?

Accepted Answer

$$\sigma_X = \sqrt{\lambda t}$$

Question 15

A life insurance company sells a $$\250$$,000$$ one-year term policy to a 49-year-old female for $$\$$530. $$The probability she survives the year is $$0.99791. $$What is the expected value to the company?

Accepted Answer

$$E(X) = 530 	imes 0.99791 + (-249470) 	imes 0.00209$$

Question 16

Which of the following is NOT a property of a Poisson process?

Accepted Answer

The probability of two or more successes in a very small interval is positive.

Discrete Random Variables, Binomial, and Poisson Distributions: Study Guide

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