7. Random Variables and Probability Distributions Free Online FlashCards

What is a random variable?

A random variable is a numerical quantity whose value is determined by the outcome of a random phenomenon.

What defines a discrete random variable?

A discrete random variable has countable possible values and is described by a probability mass function (PMF).

What defines a continuous random variable?

A continuous random variable can take any value in an interval and is described by a probability density function (PDF). Probabilities are areas over intervals.

What must all discrete probabilities sum to?

The probabilities in a discrete probability distribution must sum to 1: Σx P(X=x)=1.

How is a discrete expected value calculated?

For a discrete random variable, E(X)=Σx xP(X=x). It is the long-run average, not necessarily an outcome observed on one trial.

What is the addition rule for expected values?

E(X+Y)=E(X)+E(Y), whether or not X and Y are independent.

How are variance and standard deviation related?

Standard deviation is the square root of variance: σX=√Var(X). It uses the same units as the random variable, unlike variance.

What is the mean of a binomial distribution?

For X~Binomial(n,p), the expected value is μX=np.

What conditions define a binomial model?

A binomial model requires a fixed number of trials, two outcomes per trial, constant success probability, and independent trials.

What does a geometric distribution count?

For a geometric variable, P(X=x)=(1-p)^(x-1)p. It counts trials until the first success, where x=1,2,3,…

What are the mean and variance of a Poisson distribution?

For a Poisson distribution, E(X)=λ and Var(X)=λ. The parameter λ is the stable average event rate in a fixed interval.

What characterizes a normal distribution?

A normal distribution is continuous, symmetric, and bell-shaped; its mean determines the center and its standard deviation determines the spread.