Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable that assumes only a finite or countable number of distinct values.

Question 2

Suppose $$X is a $$random variable representing the number of heads in three coin tosses. What type of random variable is $$X$$?

Accepted Answer

Discrete random variable

Question 3

Which formula correctly represents the expected value $$E(X) of a $$discrete random variable $$X$$ with probability mass function $$p(x)$$?

Accepted Answer

$$E(X) = \sum_{x} x \cdot p(x)$$

Question 4

A hospital records the number of patients admitted each day. Which probability model is most appropriate for this scenario?

Accepted Answer

Discrete probability distribution

Question 5

If $$Y = aX + b$$, where $$a$$ and $$b$$ are constants, what is the expected value $$E(Y) in $$terms of $$E(X)$$?

Accepted Answer

$$E(Y) = aE(X) + b$$

Question 6

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

Accepted Answer

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

Question 7

A researcher measures the heights of students in centimeters. Which type of random variable is most appropriate for this data?

Accepted Answer

Continuous random variable

Question 8

If $$X$$ and $$Y$$ are independent random variables, which formula gives the variance of their sum?

Accepted Answer

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

Question 9

Which statement about the cumulative probability distribution $$F(x)$$ for a random variable $$X is $$correct?

Accepted Answer

$$F(x)$$ gives the probability that $$X is $$less than or equal to $$x$$

Question 10

A company tracks the number of defective products per day. The expected value of this random variable is 5 and the variance is 2. If the company wants to know the expected value and variance after doubling the number of products checked each day, what are the new expected value and variance?

Accepted Answer

Expected value: 10, Variance: 4

Question 11

Which of the following is NOT a property of the expected value operator $$E$$?

Accepted Answer

$$E(X^2$$) = [E(X)]^2$$

Question 12

Suppose $$X is a $$discrete random variable with $$E(X) = 4$$ and $$Var(X) = 9. $$What is $$Var(2X - 3)$$?

Accepted Answer

$$Var(2X - 3) = 36$$

Question 13

A random variable $$X$$ has $$E(X) = 10$$ and $$Var(X) = 4. If$$ $$Y = X + 5$$, what is $$E(Y)$$ and $$Var(Y)$$?

Accepted Answer

$$E(Y) = 15, Var(Y) = 4$$

Question 14

Which of the following best describes the standard deviation of a random variable $$X$$?

Accepted Answer

The square root of the variance of $$X$$

Question 15

If $$X$$ and $$Y$$ are independent random variables with $$Var(X) = 3$$ and $$Var(Y) = 7$$, what is $$Var(X + Y)$$?

Accepted Answer

$$Var(X + Y) = 10$$

Question 16

A scientist measures the concentration of a chemical in a solution, which can take any value between 0 and 1. What type of random variable is this?

Accepted Answer

Continuous random variable

Question 17

Which formula correctly expresses the addition rule for variances of independent random variables $$X$$ and $$Y$$?

Accepted Answer

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

Question 18

A random variable $$X$$ has $$E(X) = 2$$ and $$Var(X) = 5. $$What is the standard deviation of $$X$$?

Accepted Answer

$$\sqrt{5}$$

Question 19

Which of the following is true about the probability model for a random variable?

Accepted Answer

It assigns a probability to each possible value of the random variable.

Random Variables and Probability Distributions: Key Concepts and Formulas

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