Question 1

Which of the following is a discrete random variable?

Accepted Answer

The number of heads obtained when tossing a coin three times.

Question 2

A probability distribution for a discrete random variable $$X is $$given by $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3. $$What is the mean $$\mu_X of$$ $$X$$?

Accepted Answer

$$\mu_X = 1.3$$

Question 3

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

Accepted Answer

Each probability can be negative as long as the sum is 1.

Question 4

Suppose a random variable $$X$$ represents the number of defective bulbs in a box of 5, with $$P(0) = 0.1$$, $$P(1) = 0.3$$, $$P(2) = 0.4$$, $$P(3) = 0.1$$, $$P(4) = 0.05$$, $$P(5) = 0.05. $$What is $$P(X \geq 2)$$?

Accepted Answer

$$0.6$$

Question 5

Which of the following scenarios describes a binomial experiment?

Accepted Answer

Counting the number of students who pass an exam out of 30, where each has a 70% chance of passing.

Question 6

A binomial experiment has $$n = 10$$ trials and probability of success $$p = 0.4. $$What is the mean $$\mu$$ and standard deviation $$\sigma of $$the number of successes?

Accepted Answer

$$\mu = 4$$, $$\sigma = 1.55$$

Question 7

Which formula correctly gives the probability of exactly $$x$$ successes in $$n$$ independent binomial trials with probability $$p of $$success?

Accepted Answer

$$P(x) = n! / [x!(n-x)!] \cdot p^x \cdot (1-p$$)^{n-x}$$

Question 8

A survey finds that 35% of households have three or more cars. In a random sample of 20 households, what is the probability that exactly 5 have three or more cars? (Use the binomial formula.)

Accepted Answer

$$P(5) = 15504 \cdot (0.35)^5$ \cdot (0.65$$)^{15}$$ = 0.1272$$

Question 9

Which of the following random variables is continuous?

Accepted Answer

The time between calls to 911.

Question 10

Given the probability distribution: $$P(0) = 0.06$$, $$P(1) = 0.58$$, $$P(2) = 0.22$$, $$P(3) = 0.10$$, $$P(4) = 0.03$$, $$P(5) = 0.01$$, what is the expected value (mean) $$\mu_X$$?

Accepted Answer

$$\mu_X = 1.49$$

Question 11

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

Accepted Answer

The probability of success changes from trial to trial.

Question 12

A probability distribution has $$P(0) = 0.16$$, $$P(1) = 0.18$$, $$P(2) = 0.22$$, $$P(3) = 0.10$$, $$P(4) = 0.30$$, $$P(5) = -0.01. $$Why is this NOT a valid probability distribution?

Accepted Answer

Because one of the probabilities is negative.

Question 13

If $$X is a $$discrete random variable with mean $$\mu_X$$ and standard deviation $$\sigma_X$$, which formula gives the standard deviation?

Accepted Answer

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

Question 14

A medical test has a 90% chance of correctly detecting a disease. If 8 patients are tested, what is the probability that exactly 7 will test positive? (Use the binomial formula.)

Accepted Answer

$$P(7) = 8 \cdot (0.9)^7$ \cdot (0.1)^1$ = 0.3828$$

Question 15

In a binomial experiment with $$n = 15$$ and $$p = 0.2$$, what is the variance $$\$$sigma^2$$ of the number of successes?

Accepted Answer

$$\$$sigma^2$$ = 2.4$$

Question 16

A random variable $$X$$ has the following probability distribution: $$P(0) = 0.3$$, $$P(1) = 0.4$$, $$P(2) = 0.2$$, $$P(3) = 0.1. $$What is $$P(X \leq 1)$$?

Accepted Answer

$$0.7$$

Question 17

A study finds that 80% of residents favor building a health center. If 25 residents are chosen at random, what is the expected number who favor the health center?

Accepted Answer

$$20$$

Question 18

Which of the following best describes the difference between a discrete and a continuous random variable?

Accepted Answer

A discrete random variable has countable values, while a continuous random variable has uncountably infinite values.

Question 19

If the mean of a discrete random variable $$X is$$ $$\mu_X = 2$$ and $$\sum x^2$ P(x) = 6$$, what is the standard deviation $$\sigma_X$$?

Accepted Answer

$$\sigma_X = \sqrt{2}$$

Discrete and Continuous Random Variables, Probability Distributions, and Binomial Distributions

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