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

$$p_X(x) = \begin{cases} 1/4 & \text{if } x=0 \\ 1/2 & \text{if } x=1 \\ 1/4 & \text{if } x=2 \\ 0 & \text{otherwise} \end{cases}$$

Q: A life insurance company offers a $$\1$$,000$$ policy to a 50-year-old male for a premium of $$\$$14. $$The probability that the person dies within the year is $$0.006. $$What is the expected gain for the company?

$$E[X] = (14) \times 0.994 + (-986) \times 0.006 = 5$$

Q: Let $$X be a $$discrete random variable with $$E[X] = 2$$ and $$Var(X) = 3. $$What is $$Var(5X - 4)$$?

$$Var(5X - 4) = 25 \times 3 = 75$$

Q: Suppose $$X is a $$discrete random variable with PMF $$p_X(1) = 1/3$$, $$p_X(2) = 2/3. $$What is $$E[3X - 5]$$?

$$E[3X - 5] = 3 \times E[X] - 5 = 3 \times (1 \times 1/3 + 2 \times 2/3) - 5 = 45$$

Q: Suppose $$X is a $$discrete random variable with PMF $$p_X(1) = 1/3$$, $$p_X(2) = 2/3. $$What is $$E[X^2$]$$?

$$E[X^2$] = 1^2$ \times 1/3 + 2^2$ \times 2/3 = 1/3 + 8/3 = 3$$

Question 1

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

Accepted Answer

$$p_X(x) = \begin{cases} 1/4 & 	ext{if } x=0 \ 1/2 & 	ext{if } x=1 \ 1/4 & 	ext{if } x=2 \ 0 & 	ext{otherwise} \end{cases}$$

Question 2

Which of the following is NOT a property of a valid probability mass function (PMF) $$p(x)$$ for a discrete random variable $$X$$?

Accepted Answer

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

Question 3

Let $$X be a $$discrete random variable with PMF $$p_X(x) = c(x-1)$$ for $$x = 2, 3, 4. $$What is the value of the normalizing constant $$c$$?

Accepted Answer

$$c = 1/3$$

Question 4

Suppose a random variable $$X$$ has the following PMF: $$p_X(0) = 0.2$$, $$p_X(1) = 0.3$$, $$p_X(2) = 0.4$$, $$p_X(3) = 0.1. $$What is the cumulative distribution function (CDF) $$F_X(2)$$?

Accepted Answer

$$F_X(2) = 0.9$$

Question 5

A life insurance company offers a $$\1$$,000$$ policy to a 50-year-old male for a premium of $$\$$14. $$The probability that the person dies within the year is $$0.006. $$What is the expected gain for the company?

Accepted Answer

$$E[X] = (14) 	imes 0.994 + (-986) 	imes 0.006 = 5$$

Question 6

Let $$X be a $$discrete random variable with $$E[X] = 2$$ and $$Var(X) = 3. $$What is $$Var(5X - 4)$$?

Accepted Answer

$$Var(5X - 4) = 25 	imes 3 = 75$$

Question 7

Suppose $$X is a $$discrete random variable with PMF $$p_X(1) = 1/3$$, $$p_X(2) = 2/3. $$What is $$E[3X - 5]$$?

Accepted Answer

$$E[3X - 5] = 3 	imes E[X] - 5 = 3 	imes (1 	imes 1/3 + 2 	imes 2/3) - 5 = 45$$

Question 8

A company tests light bulbs and records their lifetimes (in years) as a discrete random variable $$X$$ with PMF: $$p(1) = 0.1$$, $$p(2) = 0.2$$, $$p(3) = 0.3$$, $$p(4) = 0.25$$, $$p(5) = 0.15. $$What is the conditional probability $$P(X=5 \mid X \u003e 3)$$?

Accepted Answer

$$P(X=5 \mid X \u003e 3) = \frac{0.15}{0.25 + 0.15} = 0.375$$

Question 9

Let $$X be a $$discrete random variable with PMF $$p_X(0) = 0.2$$, $$p_X(1) = 0.3$$, $$p_X(2) = 0.4$$, $$p_X(3) = 0.1. $$What is $$Var(X)$$?

Accepted Answer

$$Var(X) = E[X^2$] - (E[X])^2$ = 2.2 - 1.44 = 0.76$$

Question 10

Which of the following statements about the cumulative distribution function (CDF) $$F_X(x)$$ for a discrete random variable $$X is $$TRUE?

Accepted Answer

$$F_X(x) is a $$step function that increases at each possible value of $$X.$$

Question 11

Suppose $$X$$ and $$Y$$ are independent discrete random variables. Which of the following is TRUE about their variances?

Accepted Answer

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

Question 12

Let $$X be a $$discrete random variable with $$E[X] = 3. $$What is $$E[2X + 5]$$?

Accepted Answer

$$E[2X + 5] = 2E[X] + 5 = 11$$

Question 13

A random variable $$Y is $$defined as the number of flips of a fair coin until the first head appears. What is the PMF of $$Y$$?

Accepted Answer

$$p_Y(y) = \frac{1}{2^y}$$ for $$y = 1, 2, 3, \ldots$$

Question 14

Suppose $$X is a $$discrete random variable with PMF $$p_X(1) = 1/3$$, $$p_X(2) = 2/3. $$What is $$E[X^2$]$$?

Accepted Answer

$$E[X^2$] = 1^2$ 	imes 1/3 + 2^2$ 	imes 2/3 = 1/3 + 8/3 = 3$$

Question 15

If $$X is a $$discrete random variable and $$a$$ and $$b$$ are constants, which of the following is the correct formula for $$Var(aX + b)$$?

Accepted Answer

$$Var(aX + b) = a^2$ Var(X)$$

Question 16

A discrete random variable $$X$$ has PMF $$p_X(0) = 0.2$$, $$p_X(1) = 0.3$$, $$p_X(2) = 0.4$$, $$p_X(3) = 0.1. $$What is the conditional expectation $$E[X \mid X \geq 2]$$?

Accepted Answer

$$E[X \mid X \geq 2] = \frac{2 	imes 0.4 + 3 	imes 0.1}{0.4 + 0.1} = 2.2$$

Question 17

Which of the following best describes the difference between a probability mass function (PMF) and a cumulative distribution function (CDF) for a discrete random variable?

Accepted Answer

The PMF gives the probability at each value, while the CDF gives the probability at or below a given value.

4.2 Discrete Random Variables and Probability Distributions

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