Q: Which of the following must be true for a function $$f(x) to be a $$valid probability mass function (pmf) for a discrete random variable $$X$$?

$$0 \leq f(x) \leq 1$$ \text{ for all } x$$ and $$\sum f(x) = 1$

Q: If $$Y = 3X$$ and $$Var(X) = 0.81$$, what is $$Var(Y)$$?

$$Var(Y) = 9 \times Var(X) = 7.29$$

Question 1

Which of the following is the best definition of a random variable?

Accepted Answer

A numerical measure of the outcome from a probability experiment, determined by chance.

Question 2

Suppose you toss a fair coin 3 times and let $$X be $$the number of heads. What are the possible values of $$X$$?

Accepted Answer

$$0, 1, 2, 3$$

Question 3

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

Accepted Answer

Length of a metal rod.

Question 4

Which of the following must be true for a function $$f(x) to be a $$valid probability mass function (pmf) for a discrete random variable $$X$$?

Accepted Answer

$$0 \leq f(x) \leq 1$$ 	ext{ for all } x$$ and $$\sum f(x) = 1$

Question 5

Given the following probability distribution for $$X ($$number of movies streamed per month): $$P(0)=0.06$$, $$P(1)=0.58$$, $$P(2)=0.22$$, $$P(3)=0.10$$, $$P(4)=0.03$$, $$P(5)=0.01. $$What is the expected value (mean) of $$X$$?

Accepted Answer

$$1.57$$

Question 6

Which of the following tables cannot represent a probability distribution for a discrete random variable $$X$$?

Accepted Answer

A table where one of the probabilities is negative.

Question 7

A basketball player shoots three free throws. The experiment is repeated 100 times. The mean number of free throws made per experiment is calculated. As the number of repetitions increases, what does the mean approach?

Accepted Answer

The mean of the random variable $$X$$ representing the number of free throws made.

Question 8

A life insurance company sells a $$1-$$year term policy for $$\350$ to an 18-year-old male. If the probability the client survives the year is 0.999$ and the payout upon death is $$\$$250,000$$, what is the expected value of the policy to the company?

Accepted Answer

$$\100$

Question 9

Which formula correctly gives the mean (expected value) $$\mu_X of a $$discrete random variable $$X$$ with probability mass function $$P(x)$$?

Accepted Answer

$$\mu_X = \sum [x \cdot P(x)]$$

Question 10

Which of the following statements about the mean of a discrete random variable is correct?

Accepted Answer

The mean represents the average behavior and is not necessarily a possible value of the random variable.

Question 11

The variance of a discrete random variable $$X is $$given by which of the following formulas?

Accepted Answer

$$\$$sigma^2$$ = \sum (x - \mu)^2$ f(x)$$

Question 12

Suppose $$X is $$the number of heads and $$Y is $$the number of tails in 4 tosses of a fair coin. What is $$Var(X+Y)$$?

Accepted Answer

$$0$$

Question 13

Which of the following is NOT a property of a probability mass function (pmf) for a discrete random variable?

Accepted Answer

The sum of all probabilities must be less than 1.

Question 14

A probability histogram is best described as:

Accepted Answer

A histogram where the horizontal axis is the value of the random variable and the vertical axis is the probability of that value.

Question 15

If $$Y = 3X$$ and $$Var(X) = 0.81$$, what is $$Var(Y)$$?

Accepted Answer

$$Var(Y) = 9 	imes Var(X) = 7.29$$

Question 16

Which of the following is a shortcut formula for the variance of a discrete random variable $$X$$?

Accepted Answer

$$\$$sigma^2$$ = \sum x^2$ f(x) - \mu^2$

Question 17

Which of the following statements about the variance of the sum of two random variables $$X$$ and $$Y is $$correct?

Accepted Answer

$$Var(X+Y) is $$not always equal to $$Var(X) + Var(Y).$$

Question 18

A student claims that the mean of a discrete random variable must always be a value that the variable can actually take. Is this claim correct?

Accepted Answer

No, the mean represents the average behavior and may not be a possible value of the random variable.

Question 19

Suppose $$X is a $$discrete random variable with $$E(X) = 2.5$$ and $$Var(X) = 0.81. $$What is the standard deviation of $$X$$?

Accepted Answer

$$0.9$$

Random Variables and Probability Distributions: Study Notes

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