Chapter 3 Flashcards
What is a random variable?
A function that assigns numerical values to outcomes of a random experiment.
What is an example of a discrete random variable?
The number of heads in three coin tosses.
What makes a random variable discrete?
It has a countable range of possible values.
What is the Probability Mass Function (PMF)?
A function that gives the probability of each value a discrete random variable can take.
What is an example of a PMF?
For a fair die, P(X=1) = 1/6, P(X=2) = 1/6, etc.
What is the sum of probabilities in a PMF?
The total probability must sum to 1.
What is an independent random variable?
Two variables are independent if knowing one gives no information about the other.
What is the formula for independence of two discrete random variables?
P(X=x, Y=y) = P(X=x)P(Y=y) for all values of X and Y.
What is the Bernoulli distribution?
A distribution with two possible outcomes, success (1) and failure (0), with probability p and 1-p.
What is an example of a Bernoulli distribution?
Flipping a coin where heads = 1 and tails = 0.
What is the Binomial distribution?
A distribution modeling the number of successes in n independent Bernoulli trials.
What are the parameters of a Binomial distribution?
n (number of trials) and p (probability of success per trial).
What is the formula for the Binomial PMF?
P(X=k) = (n choose k) * p^k * (1-p)^(n-k).
What is the Negative Binomial (Pascal) distribution?
A distribution that models the number of trials needed to observe m successes.
What is the Poisson distribution used for?
Modeling the number of events occurring in a fixed interval of time or space.
What is the Poisson PMF formula?
P(X=k) = (e^(-λ) * λ^k) / k!.
What is the relationship between the Binomial and Poisson distributions?
Poisson is a limiting case of the Binomial when n is large and p is small.
What is the Cumulative Distribution Function (CDF)?
A function that gives the probability that a random variable is less than or equal to a value.
What is the expectation (expected value) of a random variable?
The long-term average value of the variable over many trials.
What is the formula for expectation?
E(X) = Σ x * P(X=x).
What is an example of expectation?
For a fair die, E(X) = (1+2+3+4+5+6)/6 = 3.5.
What is the variance of a random variable?
A measure of how spread out the values are around the mean.
What is the formula for variance?
Var(X) = E[(X - E(X))^2] = E(X^2) - (E(X))^2.
What does a higher variance indicate?
The values of X are more spread out from the mean.
What is the standard deviation?
The square root of the variance, measuring spread in the same units as X.
What is the Law of the Unconscious Statistician (LOTUS)?
A theorem stating E[g(X)] = Σ g(x) * P(X=x).
What is an example of applying LOTUS?
If X is a die roll, E(X^2) = Σ x^2 * P(X=x) = (1^2+2^2+…+6^2)/6.
What is the sum of independent random variables?
E(X+Y) = E(X) + E(Y) and Var(X+Y) = Var(X) + Var(Y) if X and Y are independent.
What is an application of the Poisson distribution?
Modeling the number of customers arriving at a store per hour.
What is a key property of the Binomial distribution?
If X~Binomial(n, p) and Y~Binomial(m, p), then X+Y ~ Binomial(n+m, p).
