Q: A random variable $$X$$ has the following probability distribution: $$P(10) = 0.6$$, $$P(20) = 0.3$$, $$P(30) = 0.1. $$What is the variance of $$X$$?

$$\text{Var}(X) = (10-14)^2$ \times 0.6 + (20-14)^2$ \times 0.3 + (30-14)^2$ \times 0.1$$

Question 1

A real estate agent in La Quinta, California, is analyzing the number of bedrooms in homes for sale. The probability distribution for the number of bedrooms $$x is $$given as follows: $$P(1) = 0.10$$, $$P(2) = 0.30$$, $$P(3) = 0.40$$, $$P(4) = 0.10$$, $$P(5) = 0.05$$, $$P(6) = 0.05. $$What is the expected number of bedrooms in a randomly selected home?

Accepted Answer

$$E(x) = 2.85$$

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

A company inspects 20 components from a shipment with a defect rate of $$0.05. $$The acceptance plan is to accept the shipment if fewer than 1 defect is found. What is the probability of accepting a good shipment?

Accepted Answer

$$0.7358$$

Question 4

Suppose you pay $3 to play a game where you roll a fair six-sided die and win an amount equal to the number rolled in dollars. What is your expected profit per game?

Accepted Answer

$$0.50$$

Question 5

Which formula correctly represents the expected value $$E(x) of a $$discrete random variable $$x$$ with probability distribution $$P(x)$$?

Accepted Answer

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

Question 6

A Poisson process has an average of $$\lambda = 5$$ arrivals per 10 minutes. What is the probability that exactly 3 arrivals occur in a 10-minute period? (Use the Poisson formula.)

Accepted Answer

$$P(x=3) = \frac{5^3$ e$$^{-5}$$}{3!}$$

Question 7

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

Accepted Answer

Counting the number of defective items in a sample drawn without replacement from a finite lot.

Question 8

A company has 20 employees, 8 of whom have college degrees. If 3 employees are selected at random without replacement, what is the probability that none of the selected employees has a college degree?

Accepted Answer

$$P = \frac{\binom{12}{3}}{\binom{20}{3}}$$

Question 9

Which of the following is a characteristic of the Poisson distribution?

Accepted Answer

The probability of an event is the same for all segments.

Question 10

If $$X is a $$binomial random variable with $$n = 10$$ and $$p = 0.35$$, what is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 3.5$$

Question 11

A marketing firm surveys 8 customers, each with a probability $$p = 0.8 of $$preferring Saturday opening. What is the probability that exactly 3 customers prefer Saturday opening? (Use the binomial formula.)

Accepted Answer

$$P(x=3) = \binom{8}{3} (0.8)^5$ (0.2)^3$

Question 12

Which formula gives the standard deviation $$\sigma of a $$binomial distribution with parameters $$n$$ and $$p$$?

Accepted Answer

$$\sigma = \sqrt{np(1-p)}$$

Question 13

A random variable $$X$$ has the following probability distribution: $$P(10) = 0.6$$, $$P(20) = 0.3$$, $$P(30) = 0.1. $$What is the variance of $$X$$?

Accepted Answer

$$	ext{Var}(X) = (10-14)^2$ 	imes 0.6 + (20-14)^2$ 	imes 0.3 + (30-14)^2$ 	imes 0.1$$

Question 14

Which of the following is an example of a discrete random variable?

Accepted Answer

The number of complaints received by a company per day.

Question 15

A hypergeometric distribution is used instead of a binomial distribution when:

Accepted Answer

Sampling is without replacement from a finite population.

Question 16

Which of the following best describes the expected value in a business context?

Accepted Answer

The long-run average outcome if the experiment is repeated many times.

Question 17

A bank wants to minimize the probability of accepting a bad shipment (defect rate $$= 0.10) $$using a sample of $$n = 20$$ and accepting if $$x \u003c 1$$ defect. What is the probability of accepting a bad shipment?

Accepted Answer

$$0.3917$$

Question 18

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

Accepted Answer

The probability of an outcome is different for each segment.

Discrete Probability Distributions: Study Notes for Business Statistics

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