Q: A company sells a $$10,000$$ one-year term insurance policy for an annual premium of $$290. $$The probability of a claim (death) is $$0.001. $$What is the expected gain for the company per policy? (Refer to the table: Gain if customer lives = $$290$$, Gain if customer dies = $$-9,710)$$

$$E(X) = 290 \times 0.999 + (-9,710) \times 0.001 = 280.29$$

Q: A paint manufacturer wants to reward production above the 90th percentile. If daily production is normally distributed with mean $$100,000$$ gallons and standard deviation $$10,000$$ gallons, what production level corresponds to the 90th percentile? (Use $$z_{0.90} \approx 1.28)$$

$$100,000 + 1.28 \times 10,000 = 112,800$$

Question 1

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

Accepted Answer

It can only assume countable values, such as 0, 1, 2, ...

Question 2

Suppose the number of customers arriving at a bank in an hour is modeled as a random variable. Which type of random variable is this, and why?

Accepted Answer

Discrete, because the number of customers is countable.

Question 3

Given the following probability distribution for a discrete random variable $$X$$:

$$P(X=0) = 0.2$$

$$P(X=1) = 0.5$$

$$P(X=2) = 0.3$$

What is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 1.3$$

Question 4

A company sells a $$10,000$$ one-year term insurance policy for an annual premium of $$290. $$The probability of a claim (death) is $$0.001. $$What is the expected gain for the company per policy?

(Refer to the table: Gain if customer lives = $$290$$, Gain if customer dies = $$-9,710)$$

Accepted Answer

$$E(X) = 290 	imes 0.999 + (-9,710) 	imes 0.001 = 280.29$$

Question 5

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

Accepted Answer

The probability of success changes from trial to trial.

Question 6

A machine produces 10% defective items. If 5 items are selected at random, what is the probability that exactly 2 are defective? (Use the binomial formula: $$P(X = x) = \binom{n}{x} p^x (1-p$$)^{n-x}$$, with n=5$, x=2$, p=0.1$)

Accepted Answer

$$P(X=2) = \binom{5}{2} (0.1)^2$ (0.9)^3$ = 0.0729$$

Question 7

Which of the following best describes the shape of a normal distribution?

Accepted Answer

Bell-shaped and symmetric

Question 8

If $$X is $$normally distributed with mean $$\mu = 27$$ and standard deviation $$\sigma = 3$$, what is the $$z-$$score for $$x = 20$$?

Accepted Answer

$$z = \frac{20 - 27}{3} = -2.33$$

Question 9

For a standard normal distribution, what is the probability that $$z is $$between 0 and 1.23? (Use the standard normal table)

Accepted Answer

0.3907

Question 10

A cell phone's battery life is normally distributed with mean $$\mu = 10$$ hours and standard deviation $$\sigma = 1.5$$ hours. What is the probability that a randomly selected phone lasts between 8 and 12 hours? (Find $$z-$$scores for 8 and 12, then use the standard normal table)

Accepted Answer

Approximately 0.8176

Question 11

Which of the following is a property of the normal distribution?

Accepted Answer

The mean, median, and mode are all equal.

Question 12

If a random variable $$X is $$continuous, which of the following statements is true?

Accepted Answer

The probability that $$X$$ takes any exact value is zero.

Question 13

A hotel guest satisfaction study finds that 26% of guests had better-than-expected sleep, and 83% of those would return. What is the probability that a randomly selected guest both had better-than-expected sleep and would return?

Accepted Answer

$$0.26 	imes 0.83 = 0.2158$$

Question 14

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

Accepted Answer

The weight of a T-bone steak

Question 15

The variance of a discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$p_i is $$given by:

Accepted Answer

$$\sum (x_i - \mu)^2$ p_i$$

Question 16

If $$X is a $$binomial random variable with $$n=10$$ and $$p=0.22$$, what is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 2.2$$

Question 17

A paint manufacturer wants to reward production above the 90th percentile. If daily production is normally distributed with mean $$100,000$$ gallons and standard deviation $$10,000$$ gallons, what production level corresponds to the 90th percentile? (Use $$z_{0.90} \approx 1.28)$$

Accepted Answer

$$100,000 + 1.28 	imes 10,000 = 112,800$$

Question 18

Which of the following statements about the standard normal distribution is FALSE?

Accepted Answer

It is skewed to the right.

Random Variables and Probability Distributions: Study Notes

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