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 $$E(X)$$ for 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 is appropriate when:

Accepted Answer

Each trial has exactly two possible outcomes and the probability of success is constant.

Question 5

In a binomial experiment with $$n = 10$$ trials and probability of success $$p = 0.3$$, what is the expected number of successes?

Accepted Answer

$$E(X) = n \cdot p = 10 \cdot 0.3 = 3$$

Question 6

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

Accepted Answer

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

Question 7

A hypergeometric distribution is used when:

Accepted Answer

Sampling is done without replacement from a finite population.

Question 8

Given a population of $$N = 50$$ employees, $$K = 10$$ are from finance, and $$n = 8$$ are selected at random without replacement. What is the expected number of finance employees selected?

Accepted Answer

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

Question 9

Which formula represents the probability mass function (PMF) for the hypergeometric distribution?

Accepted Answer

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

Question 10

Which of the following is NOT a property of a binomial experiment?

Accepted Answer

The probability of success changes from trial to trial.

Question 11

If $$X is a $$discrete random variable with possible values $$x_1$$, $$x_2$$, ..., x_n$$ and probabilities P(X = x_i$$)$$, what must be true about the sum of all probabilities?

Accepted Answer

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

Question 12

Which scenario best fits the use of a hypergeometric distribution?

Accepted Answer

Drawing cards from a deck without replacement to count the number of aces.

Question 13

What is the variance formula for 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 14

Which of the following is TRUE about continuous random variables?

Accepted Answer

Probabilities are calculated using areas under the curve over intervals.

Question 15

A student flips a coin 5 times. What is the probability of getting exactly 3 heads? (Assume $$p = 0.5$$ for heads.)

Accepted Answer

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

Question 16

Which of the following best describes the difference between binomial and hypergeometric distributions?

Accepted Answer

Binomial is for sampling with replacement; hypergeometric is for sampling without replacement.

Question 17

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

Accepted Answer

Values are usually close to the mean.

Question 18

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

Accepted Answer

Probabilities can be negative.

Discrete and Continuous Probability Distributions: Binomial & Hypergeometric

Study Guide - Practice Questions

Study Guide - Flashcards