Question 1

Which of the following statements best describes a discrete random variable?

Accepted Answer

A variable that can only take specific, separate values, such as whole numbers.

Question 2

Suppose you roll a fair six-sided die. What is the expected value $$E(X) of $$the outcome?

Accepted Answer

$$E(X) = \frac{1+2+3+4+5+6}{6} = 3.5$$

Question 3

Which formula correctly calculates the expected value of a discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$P(X = x_i)$$?

Accepted Answer

$$E(X) = \sum_{i=1}^{n} x_i \cdot P(X = x_i)$$

Question 4

A binomial distribution requires which of the following conditions?

Accepted Answer

Trials must be independent and each trial has only two possible outcomes.

Question 5

In a binomial experiment with $$n$$ trials and probability of success $$p$$, what is the formula for the mean (expected value) of the number of successes?

Accepted Answer

$$\mu = n \cdot p$$

Question 6

Which formula gives the probability of exactly $$x$$ successes in $$n$$ independent binomial trials, each with probability $$p of $$success?

Accepted Answer

$$P(X = x) = \binom{n}{x} p^x (1-p$$)^{n-x}$$

Question 7

A company has 50 employees, 10 of whom are in finance. If 8 employees are randomly selected without replacement, what is the expected number of finance employees selected?

Accepted Answer

$$E(X) = 8 \cdot \frac{10}{50} = 1.6$$

Question 8

Which scenario is best modeled by a hypergeometric distribution?

Accepted Answer

Counting the number of defective items in a sample drawn without replacement from a batch.

Question 9

What is the main difference between a binomial and a hypergeometric distribution?

Accepted Answer

Binomial distribution is used for sampling with replacement, hypergeometric for sampling without replacement.

Question 10

Which formula correctly gives the variance of a discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$P(X = x_i)$$?

Accepted Answer

$$Var(X) = \sum_{i=1}^{n} (x_i - E(X))^2$ \cdot P(X = x_i)$$

Question 11

If you flip a fair coin 5 times, what is the probability of getting exactly 3 heads?

Accepted Answer

$$P(X=3) = \binom{5}{3} (0.5)^3$ (0.5$$)^{2}$$ = 10 \cdot 0.125 \cdot 0.25 = 0.3125$$

Question 12

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

Accepted Answer

The sum of all probabilities can be greater than 1.

Question 13

A continuous random variable differs from a discrete random variable in that:

Accepted Answer

Its probabilities are found using areas under a curve over intervals.

Question 14

Which formula gives the probability mass function (PMF) for the hypergeometric distribution, where $$N is $$population size, $$K is $$number of items of interest, $$n is $$number of draws, and $$x is $$number of successes?

Accepted Answer

$$P(X = x) = \frac{\binom{K}{x} \binom{N-K}{n-x}}{\binom{N}{n}}$$

Question 15

If the variance of a discrete random variable is small, what does this indicate about the spread of its values?

Accepted Answer

Values are usually close to the mean.

Question 16

Which of the following is a valid example of a discrete random variable?

Accepted Answer

The number of defective bulbs in a box of 100.

Question 17

In a binomial experiment with $$n=8$$ and $$p=0.25$$, what is the expected number of successes?

Accepted Answer

$$E(X) = 8 \cdot 0.25 = 2$$

Question 18

Which of the following best describes the probability of a continuous random variable taking one exact value?

Accepted Answer

It is always zero.

Discrete and Continuous Probability Distributions: Binomial and Hypergeometric

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