Q: Given a discrete random variable X with possible values 0, 1, 2, and 3, and corresponding probabilities 0.1, 0.3, 0.4, and 0.2, what is the mean of X?

$$0.1 \times 0 + 0.3 \times 1 + 0.4 \times 2 + 0.2 \times 3 = 1.7$$

Q: A life insurance company sells a $$250,000 policy to an 18-year-old male for 350. $$The probability the client survives the year is 0.999. What is the expected value of the policy to the company?

$$0.999 \times 350 + 0.001 \times (-249650) = 100$$

Q: For a binomial random variable with $$n = 300$$ and $$p = 0.72$$, what is the mean and standard deviation?

Mean: $$\mu = 300 \times 0.72 = 216$$; Standard deviation: $$\sigma = \sqrt{300 \times 0.72 \times 0.28} \approx 7.74$$

Question 1

Which of the following is a characteristic of a discrete random variable?

Accepted Answer

It has a finite or countable number of possible values.

Question 2

Suppose X is the number of cars passing through a drive-through in the next hour. Which type of random variable is X?

Accepted Answer

Discrete, because the number of cars can be counted.

Question 3

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

Accepted Answer

Each probability must be greater than 1.

Question 4

Given a discrete random variable X with possible values 0, 1, 2, and 3, and corresponding probabilities 0.1, 0.3, 0.4, and 0.2, what is the mean of X?

Accepted Answer

$$0.1 	imes 0 + 0.3 	imes 1 + 0.4 	imes 2 + 0.2 	imes 3 = 1.7$$

Question 5

A life insurance company sells a $$250,000 policy to an 18-year-old male for 350. $$The probability the client survives the year is 0.999. What is the expected value of the policy to the company?

Accepted Answer

$$0.999 	imes 350 + 0.001 	imes (-249650) = 100$$

Question 6

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

Accepted Answer

$$\sqrt{\sum (x - \mu)^2$ P(x)}$$

Question 7

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

Accepted Answer

The probability of success changes from trial to trial.

Question 8

A basketball player makes 80% of free throws. If she shoots 3 times, what is the probability she makes exactly 2 shots? (Use the binomial formula.)

Accepted Answer

$$\binom{3}{2} (0.8)^2$ (0.2)^1$ = 0.384$$

Question 9

In a binomial experiment with $$n = 15$$ and $$p = 0.65$$, what is the probability that between 8 and 10 (inclusive) are successes? (Refer to the R output in the image.)

Accepted Answer

$$0.5348254$$

Question 10

For a binomial random variable with $$n = 300$$ and $$p = 0.72$$, what is the mean and standard deviation?

Accepted Answer

Mean: $$\mu = 300 	imes 0.72 = 216$$; Standard deviation: $$\sigma = \sqrt{300 	imes 0.72 	imes 0.28} \approx 7.74$$

Question 11

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

Accepted Answer

The number of cars arriving at a drive-through in a 10-minute interval.

Question 12

A McDonald’s manager knows cars arrive at a rate of 2 per minute. What is the probability that exactly 6 cars arrive in 5 minutes? (Use the Poisson formula.)

Accepted Answer

$$\frac{e$$^{-10}$$ 10^6$}{6!}$$

Question 13

If the mean number of beetles in a 5-square-foot subsection is 10, what is the standard deviation of the number of beetles in a subsection?

Accepted Answer

$$\sqrt{10}$$

Question 14

Which of the following statements about the shape of a binomial distribution is correct?

Accepted Answer

For fixed $$p$$, as $$n$$ increases, the distribution becomes more bell-shaped.

Question 15

A probability experiment consists of drawing three cards from a deck without replacement and recording the number of aces. Is this a binomial experiment?

Accepted Answer

No, because the trials are not independent.

Question 16

In a Poisson process, which of the following is NOT a required condition?

Accepted Answer

The number of successes in overlapping intervals is dependent.

Question 17

A biologist finds more than 16 beetles in a 5-square-foot subsection, where the mean is 10. According to the Poisson model, is this result unusual?

Accepted Answer

Yes, because it occurs about 3 times in 100 trials.

Discrete Probability Distributions: Study Guide for Introductory Statistics

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