Question 1

Which of the following statements best describes a discrete random variable?

Accepted Answer

It can only take values that are whole numbers or countable.

Question 2

Suppose you flip two fair coins. Let $$X be $$the number of heads. What is the probability mass function (PMF) of $$X$$?

Accepted Answer

$$p_X(x) = \frac{1}{4} if$$ $$x=0$$, $$\frac{1}{2} if$$ $$x=1$$, $$\frac{1}{4} if$$ $$x=2$$, $$0$$ otherwise.

Question 3

Which of the following is NOT a property of a valid probability mass function (PMF) $$p(x)$$?

Accepted Answer

$$\sum_x p(x) \u003c 1.$$

Question 4

A box contains three balls labeled 1, 2, and 2. If $$X is $$the number written on the ball drawn at random, what is $$P(X=2)$$?

Accepted Answer

$$\frac{2}{3}$$

Question 5

Let $$Y be $$the number of flips of a fair coin until the first head appears. What is the probability that $$Y \u003c 3$$?

Accepted Answer

$$\frac{3}{4}$$

Question 6

Given a discrete random variable $$X$$ with PMF $$p_X(x)$$, which formula gives the expected value $$E[X]$$?

Accepted Answer

$$E[X] = \sum_x x p_X(x)$$

Question 7

An insurance company offers a $$1,000 policy for a premium of 14. $$The probability of payout is $$0.006. $$What is the expected gain for the company?

Accepted Answer

$$8$$

Question 8

If $$X is a $$discrete random variable with $$E[X] = 5$$ and $$Var(X) = 4$$, what is $$Var(2X + 3)$$?

Accepted Answer

$$16$$

Question 9

Which of the following statements about expected value is TRUE?

Accepted Answer

The expected value is the long-run average of the random variable under repeated experiments.

Question 10

Suppose $$X$$ takes values in $$\{2,3,4\}$$ with $$P(X=x) = c(x-1)$$ for $$x=2,3,4. $$What is the value of $$c$$?

Accepted Answer

$$\frac{1}{3}$$

Question 11

Which of the following is the definition of the cumulative distribution function (CDF) for a discrete random variable $$X$$?

Accepted Answer

$$F_X(x) = P(X \leq x)$$

Question 12

If $$X is a $$discrete random variable with $$E[X] = 2$$ and $$E[X^2$] = 6$$, what is $$Var(X)$$?

Accepted Answer

$$4$$

Question 13

Which of the following is TRUE about the standard deviation $$sd(X) of a $$discrete random variable $$X$$?

Accepted Answer

It is the square root of the variance.

Question 14

Let $$X be $$the number of heads in two flips of a fair coin. What is $$E[X]$$?

Accepted Answer

$$1$$

Question 15

If $$p(x) is a $$PMF defined by $$p(0)=0.2$$, $$p(2)=0.4$$, $$p(3)=0.1$$, what must $$p(1) be $$for $$p(x) to be a $$valid PMF?

Accepted Answer

$$0.3$$

Question 16

Which of the following random variables is continuous?

Accepted Answer

The amount of gas in a car's tank.

Question 17

If $$X$$ and $$Y$$ are identically distributed random variables, which of the following must be TRUE?

Accepted Answer

They have the same PMF.

Question 18

Suppose $$E[X] = 3$$ and $$E[Y] = 5. $$What is $$E[2X + Y]$$?

Accepted Answer

$$11$$

Question 19

Which of the following is a correct statement about $$E[f(X)]$$ for a function $$f$$ and discrete random variable $$X$$?

Accepted Answer

$$E[f(X)] = \sum_x f(x) p_X(x)$$

Question 20

If $$Var(X) = 0$$, what can you conclude about $$X$$?

Accepted Answer

X is always equal to a constant value.

4.1 Discrete Random Variables and Probability Distributions

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