Q: Given the probability model for a life insurance policy: $$P(X = 100{,}000) = \frac{1}{1000}$$, $$P(X = 50{,}000) = \frac{2}{1000}$$, $$P(X = 0) = \frac{997}{1000}$$, what is the expected annual payout?

$$E(X) = 100{,}000 \times \frac{1}{1000} + 50{,}000 \times \frac{2}{1000} + 0 \times \frac{997}{1000} = 200$$

Q: Suppose a customer at a bookstore can buy 0, 1, or 2 books with probabilities 0.2, 0.4, and 0.4, respectively. What is the expected number of books purchased?

$$E(X) = 0 \times 0.2 + 1 \times 0.4 + 2 \times 0.4 = 1.2$$

Q: A tennis player makes a successful first serve 67% of the time. In 6 serves, what is the expected number of successful serves?

$$E(X) = np = 6 \times 0.67 = 4.02$$

Q: A salesman closes a sale on 80% of his presentations. What is the probability he closes his first sale on the fourth attempt?

$$P(X = 4) = (1 - 0.8$$)^{3}$$ \times 0.8 = 0.008$$

Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable whose possible values can be listed and are based on the outcome of a random event.

Question 2

Given the probability model for a life insurance policy: $$P(X = 100{,}000) = \frac{1}{1000}$$, $$P(X = 50{,}000) = \frac{2}{1000}$$, $$P(X = 0) = \frac{997}{1000}$$, what is the expected annual payout?

Accepted Answer

$$E(X) = 100{,}000 	imes \frac{1}{1000} + 50{,}000 	imes \frac{2}{1000} + 0 	imes \frac{997}{1000} = 200$$

Question 3

Suppose a customer at a bookstore can buy 0, 1, or 2 books with probabilities 0.2, 0.4, and 0.4, respectively. What is the expected number of books purchased?

Accepted Answer

$$E(X) = 0 	imes 0.2 + 1 	imes 0.4 + 2 	imes 0.4 = 1.2$$

Question 4

If $$X is a $$random variable with $$E(X) = 10$$ and $$Var(X) = 25$$, what is $$E(2X + 3)$$ and $$Var(2X + 3)$$?

Accepted Answer

$$E(2X + 3) = 23$$, $$Var(2X + 3) = 100$$

Question 5

A tennis player makes a successful first serve 67% of the time. In 6 serves, what is the expected number of successful serves?

Accepted Answer

$$E(X) = np = 6 	imes 0.67 = 4.02$$

Question 6

Which probability model is most appropriate for predicting the number of purchases per minute on a website, given an average of 3 purchases per minute?

Accepted Answer

Poisson model

Question 7

A salesman closes a sale on 80% of his presentations. What is the probability he closes his first sale on the fourth attempt?

Accepted Answer

$$P(X = 4) = (1 - 0.8$$)^{3}$$ 	imes 0.8 = 0.008$$

Question 8

If $$X$$ and $$Y$$ are independent random variables with $$Var(X) = 9$$ and $$Var(Y) = 16$$, what is $$Var(X - Y)$$?

Accepted Answer

$$Var(X - Y) = Var(X) + Var(Y) = 25$$

Question 9

Which of the following is NOT a requirement for Bernoulli trials?

Accepted Answer

The probability of success changes from trial to trial.

Question 10

A random variable $$X$$ follows a Poisson distribution with mean $$\lambda = 3. $$What is the probability that no events occur in a given interval?

Accepted Answer

$$P(X = 0) = \frac{e$$^{-3}$$ 3$$^{0}$$}{0!} = e$$^{-3}$$

Question 11

If $$X is a $$discrete uniform random variable on $$\{1, 2, 3, 4, 5, 6\}$$, what is $$P(X = 4)$$?

Accepted Answer

$$P(X = 4) = \frac{1}{6}$$

Question 12

A cable provider randomly selects a phone number from 10,000 possible numbers. What is the probability the number ends in 000?

Accepted Answer

$$P = \frac{1}{1000} = 0.001$$

Question 13

Which formula gives the standard deviation of a binomial random variable with parameters $$n$$ and $$p$$?

Accepted Answer

$$\sigma = \sqrt{npq}$$,\ where\ q = 1 - p$

Question 14

If the payout amounts in an insurance policy are doubled, how does this affect the expected value and variance?

Accepted Answer

The expected value is doubled and the variance is quadrupled.

Question 15

Which probability model is most appropriate for predicting the number of trials until the first success in a sequence of independent Bernoulli trials?

Accepted Answer

Geometric model

Question 16

A website manager wants to know the probability that at least one purchase is made in a minute, given purchases follow a Poisson process with $$\lambda = 3. $$What is this probability?

Accepted Answer

$$P(X \geq 1) = 1 - e$$^{-3}$$

Question 17

Which of the following statements about variances is correct?

Accepted Answer

Variances of independent random variables add, even when considering their difference.

Question 18

Which of the following is a property of the expected value when adding a constant $$c to a $$random variable $$X$$?

Accepted Answer

$$E(X + c) = E(X) + c$$

Random Variables and Probability Models – Study Notes

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