Random variables and distributions
Discrete and continuous models, their parameters, and the named distributions the course uses.
20 terms
Bernoulli trialA Bernoulli trial is one random trial with exactly two outcomes, success or failure, and a fixed probability p of success.Binomial coefficientThe binomial coefficient, read n choose k, counts how many ways k successes can occur among n trials when the order does not matter.Binomial distributionThe binomial distribution gives the probability of a set number of successes in a fixed number of independent trials with a constant success probability.Continuous random variableA continuous random variable can take any value in an interval, so its probabilities come from area under a density curve, not from single points.Cumulative distributionA cumulative distribution gives the probability that a random variable is at most a given value, adding probabilities from the smallest value upward.Discrete random variableA discrete random variable takes a countable set of separate values, each with its own probability, and those probabilities add to 1.Empirical ruleThe empirical rule says that in a normal distribution, about 68, 95, and 99.7 percent of values fall within 1, 2, and 3 standard deviations of the mean.Expected valueThe expected value of a random variable is its long-run average, found by multiplying each value by its probability and adding the products.Geometric distributionThe geometric distribution models the number of independent trials up to and including the first success, when each trial has success probability p.Linear transformation of a random variableA linear transformation rescales and shifts a random variable, moving its mean by the same rule while changing its standard deviation only by the multiplier.Mean of a random variableThe mean of a random variable, written mu-X, is its long-run average: multiply each value by its probability and add all the products.Normal approximation to the binomialThe normal approximation to the binomial models a binomial count with a normal curve of mean np and standard deviation the square root of np(1-p).Normal distributionThe normal distribution is a continuous bell-shaped model in which probability is area under a curve fixed entirely by the mean and the standard deviation.Poisson distributionThe Poisson distribution models the count of events in a fixed interval when events occur independently at a constant average rate.Probability distributionA probability distribution is the rule assigning probability to the values of a random variable, with the probabilities over all values totalling 1.Probability histogramA probability histogram is a bar graph of a discrete probability distribution in which each bar's height is the probability of that value.Random variableA random variable assigns a numerical value to each outcome of a chance process, so its value is determined by the result of a random event.Standard deviation of a random variableThe standard deviation of a random variable, sigma-X, is the typical distance of its values from the mean mu-X over the long run.Standard normal distributionThe standard normal distribution is the normal distribution with mean 0 and standard deviation 1, on which z-scores are read.t-distributionThe t-distribution is a symmetric, bell-shaped curve with heavier tails than the normal, used for inference about a mean when the population SD is unknown.