Question 1

Suppose $$X is a $$discrete random variable with probability mass function $$P(X = 1) = 0.4$$, $$P(X = 2) = 0.2$$, $$P(X = 3) = 0.3$$, $$P(X = 4) = 0.1. $$What is the expected value $$E[X]$$?

Accepted Answer

$$E[X] = 2.5$$

Question 2

Given the same random variable $$X as $$above, what is the second moment $$E[X^2$]$$?

Accepted Answer

$$E[X^2$] = 7.0$$

Question 3

For the random variable $$X$$ above, what is the first central moment $$E[X - \mu]$$?

Accepted Answer

$$E[X - \mu] = 0$$

Question 4

For the same $$X$$, what is the variance $$Var(X)$$?

Accepted Answer

$$Var(X) = 1.75$$

Question 5

If $$Y = g(X) is a $$transformation of a discrete random variable $$X$$, what is the formula for the expected value of $$Y$$?

Accepted Answer

$$E[Y] = \sum_i g(x_i)P(X = x_i)$$

Question 6

Suppose $$E(X) = 5$$ and $$Y = 3X + 2. $$What is $$E(Y)$$?

Accepted Answer

$$E(Y) = 17$$

Question 7

Let $$X$$ have the following probability distribution: $$x = -3, 6, 9$$ with $$f(x) = \frac{1}{12}, \frac{1}{2}, \frac{5}{12}$$, respectively. What is $$E[(3X + 3)^2$]$$?

Accepted Answer

$$E[(3X + 3)^2$] = 243$$

Question 8

The random variable $$X$$ represents the number of errors per 100 lines of software code, with the following distribution: $$x = 2, 3, 4, 5, 6$$ and $$f(x) = 0.04, 0.01, 0.40, 0.25, 0.30. $$What is the variance of $$X$$?

Accepted Answer

$$Var(X) = 1.16$$

Question 9

Let $$X$$ have the following probability distribution: $$x = -2, 4, 6$$ with $$f(x) = \frac{1}{3}, \frac{1}{12}, \frac{7}{12}. $$Find $$E(X).$$

Accepted Answer

$$E(X) = 4.5$$

Question 10

For the same $$X as $$above, what is $$E(X^2$$)$$?

Accepted Answer

$$E(X^2) = 25.5$

Question 11

Using your answers from the previous question, what is $$E[(3X + 1)^2$]$$?

Accepted Answer

$$E[(3X + 1)^2$] = 139.5$$

Question 12

Which of the following is the correct formula for the $$kth $$raw moment of a discrete random variable $$X$$?

Accepted Answer

$$E[X^k] = \sum_i (x_i)^k P(X = x_i)$$

Question 13

Which of the following is the correct formula for the $$kth $$central moment of a discrete random variable $$X$$?

Accepted Answer

$$E[(X - \mu)^k] = \sum_i (x_i - \mu)^k P(X = x_i)$$

Question 14

Suppose $$X$$ and $$Y$$ are two random variables and $$a, b$$ are constants. What is the expected value of $$aX + bY$$?

Accepted Answer

$$E[aX + bY] = aE[X] + bE[Y]$$

Question 15

A hospital records the number of patients arriving each hour as a discrete random variable $$X. If$$ $$E[X] = 8$$ and $$Y = 2X - 3$$, what is $$E[Y]$$?

Accepted Answer

$$E[Y] = 19$$

Question 16

A quality control engineer models the number of defective items in a batch as a discrete random variable $$X$$ with $$E[X] = 2$$ and $$Var(X) = 0.5. $$What is the expected value of $$Z = 4X + 1$$?

Accepted Answer

$$E[Z] = 9$$

Question 17

A random variable $$X$$ has $$E[X] = 10$$ and $$E[X^2$] = 120. $$What is the variance of $$X$$?

Accepted Answer

$$Var(X) = 20$$

Question 18

A discrete random variable $$X$$ has $$E[X] = 4$$ and $$Var(X) = 3. $$What is $$E[(X - 4)^2$]$$?

Accepted Answer

$$E[(X - 4)^2$] = 3$$

Discrete Random Variables: Expected Value and Moments

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