Question 1

Which of the following best describes a random variable?

Accepted Answer

A variable whose value is determined by the outcome of a random event.

Question 2

In the example where an engineer selects one resistor labeled 10 Ω and one labeled 20 Ω, and assigns the sum of their actual resistances as $$X$$, what type of random variable is $$X$$?

Accepted Answer

Discrete random variable, because the sum can only take isolated values.

Question 3

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

Accepted Answer

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

Question 4

A certain industrial process is recalibrated during a week with the following probability mass function: $$P(X=0)=0.35$$, $$P(X=1)=0.25$$, $$P(X=2)=0.20$$, $$P(X=3)=0.15$$, $$P(X=4)=0.05. $$What is the expected number of recalibrations required in a week?

Accepted Answer

$$E(X) = 1.30$$

Question 5

Which of the following is the correct formula for the variance of a discrete random variable $$X$$ with mean $$\mu$$ and probability mass function $$p(x)$$?

Accepted Answer

$$Var(X) = \sum_{x} (x - \mu)^2$ p(x)$$

Question 6

If $$X is a $$discrete random variable with $$E(X) = 1.30$$ and $$Var(X) = 1.51$$, what is the standard deviation of $$X$$?

Accepted Answer

$$SD(X) = 1.23$$

Question 7

Suppose you flip three identical fair coins. Let $$X be $$the total number of heads. What is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 1.5$$

Question 8

If $$Y = 3X + 2$$ and $$E(X) = 5$$, what is $$E(Y)$$?

Accepted Answer

$$E(Y) = 17$$

Question 9

If $$X is a $$random variable and $$Var(X) = 4$$, what is $$Var(2X - 3)$$?

Accepted Answer

$$Var(2X - 3) = 16$$

Question 10

Suppose the time it takes a customer to get and pay for seats at a ticket window is a random variable with mean $$100$$ seconds and standard deviation $$50$$ seconds. If there are two people ahead of you, what is the expected total wait time?

Accepted Answer

$$200$$ seconds

Question 11

In the previous question, what is the standard deviation of your total wait time, assuming the two customers are independent?

Accepted Answer

$$SD = \sqrt{50^2$ + 50^2$} = 70.71$$ seconds

Question 12

If $$X$$ and $$Y$$ are independent random variables with $$Var(X) = 9$$ and $$Var(Y) = 16$$, what is $$Var(X + Y)$$?

Accepted Answer

$$25$$

Question 13

A binomial random variable $$X$$ with parameters $$n$$ and $$p$$ has expected value and variance given by:

Accepted Answer

$$E(X) = np$$, $$Var(X) = np(1-p)$$

Question 14

Which of the following scenarios corresponds to a binomial model?

Accepted Answer

The number of students among a randomly surveyed group who have taken Statistics.

Question 15

A grocery store finds that 5% of its packages are not sealed properly. In a case of 24 packages, what is the probability that none are unsealed?

Accepted Answer

$$P(X=0) = (0.95$$)^{24}$$

Question 16

In a binomial experiment with $$n=12$$ and $$p=0.2$$, what is the probability that fewer than 2 invoices receive a discount?

Accepted Answer

$$P(X \u003c 2) = P(X=0) + P(X=1)$$

Question 17

If $$X$$ and $$Y$$ are random variables, which formula gives the covariance between $$X$$ and $$Y$$?

Accepted Answer

$$Cov(X, Y) = E[(X - \mu_X)(Y - \mu_Y)]$$

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

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

Random Variables and the Binomial Model: Key Concepts and Applications

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