Chapter 3

Question 1

What is a random variable?

Answer 1

A function that assigns numerical values to outcomes of a random experiment.

Question 2

What is an example of a discrete random variable?

Answer 2

The number of heads in three coin tosses.

Question 3

What makes a random variable discrete?

Answer 3

It has a countable range of possible values.

Question 4

What is the Probability Mass Function (PMF)?

Answer 4

A function that gives the probability of each value a discrete random variable can take.

Question 5

What is an example of a PMF?

Answer 5

For a fair die, P(X=1) = 1/6, P(X=2) = 1/6, etc.

Question 6

What is the sum of probabilities in a PMF?

Answer 6

The total probability must sum to 1.

Question 7

What is an independent random variable?

Answer 7

Two variables are independent if knowing one gives no information about the other.

Question 8

What is the formula for independence of two discrete random variables?

Answer 8

P(X=x, Y=y) = P(X=x)P(Y=y) for all values of X and Y.

Question 9

What is the Bernoulli distribution?

Answer 9

A distribution with two possible outcomes, success (1) and failure (0), with probability p and 1-p.

Question 10

What is an example of a Bernoulli distribution?

Answer 10

Flipping a coin where heads = 1 and tails = 0.

Question 11

What is the Binomial distribution?

Answer 11

A distribution modeling the number of successes in n independent Bernoulli trials.

Question 12

What are the parameters of a Binomial distribution?

Answer 12

n (number of trials) and p (probability of success per trial).

Question 13

What is the formula for the Binomial PMF?

Answer 13

P(X=k) = (n choose k) * p^k * (1-p)^(n-k).

Question 14

What is the Negative Binomial (Pascal) distribution?

Answer 14

A distribution that models the number of trials needed to observe m successes.

Question 15

What is the Poisson distribution used for?

Answer 15

Modeling the number of events occurring in a fixed interval of time or space.

Question 16

What is the Poisson PMF formula?

Answer 16

P(X=k) = (e^(-λ) * λ^k) / k!.

Question 17

What is the relationship between the Binomial and Poisson distributions?

Answer 17

Poisson is a limiting case of the Binomial when n is large and p is small.

Question 18

What is the Cumulative Distribution Function (CDF)?

Answer 18

A function that gives the probability that a random variable is less than or equal to a value.

Question 19

What is the expectation (expected value) of a random variable?

Answer 19

The long-term average value of the variable over many trials.

Question 20

What is the formula for expectation?

Answer 20

E(X) = Σ x * P(X=x).

Question 21

What is an example of expectation?

Answer 21

For a fair die, E(X) = (1+2+3+4+5+6)/6 = 3.5.

Question 22

What is the variance of a random variable?

Answer 22

A measure of how spread out the values are around the mean.

Question 23

What is the formula for variance?

Answer 23

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

Question 24

What does a higher variance indicate?

Answer 24

The values of X are more spread out from the mean.

Question 25

What is the standard deviation?

Answer 25

The square root of the variance, measuring spread in the same units as X.

Question 26

What is the Law of the Unconscious Statistician (LOTUS)?

Answer 26

A theorem stating E[g(X)] = Σ g(x) * P(X=x).

Question 27

What is an example of applying LOTUS?

Answer 27

If X is a die roll, E(X^2) = Σ x^2 * P(X=x) = (1^2+2^2+...+6^2)/6.

Question 28

What is the sum of independent random variables?

Answer 28

E(X+Y) = E(X) + E(Y) and Var(X+Y) = Var(X) + Var(Y) if X and Y are independent.

Question 29

What is an application of the Poisson distribution?

Answer 29

Modeling the number of customers arriving at a store per hour.

Question 30

What is a key property of the Binomial distribution?

Answer 30

If X~Binomial(n, p) and Y~Binomial(m, p), then X+Y ~ Binomial(n+m, p).