Question 1

Which of the following best describes a random variable?

Accepted Answer

A numerical measurement of the outcome of a random phenomenon.

Question 2

Suppose you flip a fair coin three times. Let $$X be $$the number of heads observed. What is the probability that $$X = 2$$?

Accepted Answer

$$0.375$$

Question 3

Which of the following is a requirement for the probability distribution of a discrete random variable $$X$$?

Accepted Answer

Each probability $$P(x)$$ must be between 0 and 1, and the sum of all $$P(x)$$ must equal 1.

Question 4

In a best-of-seven series, the probability that the series lasts at least six games is $$0.625. $$Which calculation gives this result?

Accepted Answer

$$P(6) + P(7)$$

Question 5

The mean of a discrete probability distribution is given by which formula?

Accepted Answer

$$\mu = \sum x \cdot P(x)$$

Question 6

A random variable $$X$$ has possible values $$0, 1, 2$$ with probabilities $$0.2, 0.5, 0.3$$ respectively. What is the expected value of $$X$$?

Accepted Answer

$$1.1$$

Question 7

Which of the following statements about the expected value of a random variable is TRUE?

Accepted Answer

The expected value reflects the long-run average outcome over many repetitions.

Question 8

You are given $$1000 to invest. You can choose between a sure gain of 500$$, or a 0.5 chance to gain $1000 and a 0.5 chance to gain nothing. What is the expected gain for the risk-taking strategy?

Accepted Answer

$$500$$

Question 9

Which of the following best describes the standard deviation $$\sigma of a $$probability distribution?

Accepted Answer

It measures the variability from the mean.

Question 10

For a categorical variable with two categories coded as 0 and 1, what does the mean of the random variable represent?

Accepted Answer

The probability of the outcome coded as 1.

Question 11

Which of the following is an example of a continuous random variable?

Accepted Answer

The time it takes to run a marathon.

Question 12

For a continuous random variable, how is the probability that the variable falls within a particular interval determined?

Accepted Answer

By the area under the probability density curve above the interval.

Question 13

If the area under the probability density curve for commuting times less than 15 minutes is $$0.29$$, what does this represent?

Accepted Answer

The probability that a randomly selected commuting time is less than 15 minutes.

Question 14

Which of the following is NOT a property of a probability distribution for a continuous random variable?

Accepted Answer

The sum of the probabilities for all possible values equals 1.

Question 15

A discrete random variable $$X$$ has the following probability distribution: $$P(0) = 0.1$$, $$P(1) = 0.3$$, $$P(2) = 0.6. $$What is the mean $$\mu of$$ $$X$$?

Accepted Answer

$$1.5$$

Question 16

Which of the following real-world scenarios is best modeled by a discrete random variable?

Accepted Answer

The number of defective products in a batch of 100.

Question 17

Suppose a random variable $$X$$ has possible values $$1, 2, 3$$ with probabilities $$0.2, 0.5, 0.3$$ respectively. What is the probability that $$X is at $$least 2?

Accepted Answer

$$0.8$$

Question 18

Which of the following statements about randomness is correct?

Accepted Answer

Randomness allows us to specify probabilities for possible outcomes.

Question 19

A probability distribution is called a parameter when it describes which of the following?

Accepted Answer

A population distribution.

Question 20

If a random variable $$X$$ has a mean $$\mu = 10$$ and a standard deviation $$\sigma = 2$$, which of the following is true?

Accepted Answer

On average, values of $$X$$ fall about 2 units from the mean.

Probability Distributions: Discrete and Continuous Random Variables

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