Random Variables and Distributions: every key term you need (+ practice quiz)
25 flashcard terms for Probability & Statistical Inference Topic 2, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
A measurable function from the sample space to the real numbers, turning outcomes into numbers so arithmetic and calculus apply. It is a function, not a number, and the randomness lives in the outcome that is fed into it.
Discrete random variable
A random variable whose set of possible values is finite or countable, described by a probability mass function that sums to one. Counts, categories and waiting times measured in whole trials all fall in this class.
Continuous random variable
A random variable with a density whose integral over the real line is one, so every single point has probability zero. Only intervals carry positive probability, which is why endpoints do not matter for continuous events.
Probability mass function
A function assigning to each possible value of a discrete variable the probability of that exact value. It is nonnegative and its values sum to one over the support of the variable.
Probability density function
A nonnegative function whose area over an interval gives the probability of landing in that interval. Its height is not a probability and may exceed one where the distribution is tightly concentrated.
Cumulative distribution function
The probability that a variable is at most a given value. It is nondecreasing, right continuous, tends to zero and one at the extremes, and determines the distribution completely for both discrete and continuous cases.
Support of a distribution
The set of values where the mass or density is positive. Many mistakes come from ignoring the support, for instance integrating a density outside the range where it is actually defined.
Quantile function
The generalised inverse of the distribution function, returning the smallest value whose cumulative probability reaches a given level. It supplies medians, quartiles and the critical values used in interval construction.
Bernoulli distribution
The model for one trial with two outcomes coded one and zero and success probability p. It is the atom from which the binomial, geometric and negative binomial families are built by repetition.
Binomial distribution
Counts successes in a fixed number of independent trials with constant success probability. Both the fixed trial count and the constant probability matter; sampling without replacement from a small population breaks the second.
Geometric distribution
Counts trials until the first success in independent trials with constant success probability. It is the only discrete distribution with the memoryless property, so past failures do not shorten the remaining wait.
Negative binomial distribution
Counts trials needed to reach a fixed number of successes, generalising the geometric case. It is also used as an overdispersed alternative to the Poisson when the variance exceeds the mean.
Hypergeometric distribution
Counts successes when sampling without replacement from a finite population of known composition. It replaces the binomial whenever the sample is a substantial fraction of the population.
Poisson distribution
Models counts of events in a fixed window when events occur independently at a constant average rate. Its mean and variance are both the rate parameter, which gives a quick check on whether the model fits.
Poisson limit of the binomial
When the trial count grows large and the success probability shrinks with their product held roughly constant, binomial probabilities converge to Poisson probabilities. This justifies Poisson models for rare events among many opportunities.
Discrete uniform distribution
Equal probability on a finite set of values, the formal version of the equally likely outcomes model. It is the reference case for fair dice, random digits and simple random selection from a list.
Continuous uniform distribution
Constant density on an interval, so probability is proportional to interval length. It underpins simulation because applying an inverse distribution function to a uniform draw generates any desired distribution.
Exponential distribution
The continuous waiting time between events of a Poisson process, with density decaying at a constant rate. It is memoryless, so a component that has survived a long time is no more likely to fail soon.
Gamma distribution
The waiting time until a fixed number of Poisson events, with a shape and a rate parameter. It contains the exponential and chi-square families as special cases and is a common prior for positive rates.
Beta distribution
A flexible family on the interval from zero to one, governed by two positive shape parameters. It is the natural model for an unknown proportion and the conjugate partner of the binomial likelihood.
Normal distribution
The symmetric bell-shaped family determined by a mean and a variance, closed under linear transformation and under summation of independent members. Its dominance comes from the central limit theorem rather than from any law of nature.
Standard normal distribution
The normal distribution with mean zero and variance one, obtained by subtracting the mean and dividing by the standard deviation. Tabulated critical values refer to this standardised version.
Lognormal distribution
The distribution of a variable whose logarithm is normal, hence positive and right skewed. It arises when effects multiply rather than add, as with growth rates, incomes and particle sizes.
Memoryless property
The condition that the remaining wait, given survival so far, has the same distribution as the original wait. Only the geometric and exponential distributions satisfy it, which sharply limits where constant-hazard models are appropriate.
Transformation of a random variable
Deriving the distribution of a function of a variable, either by tracking the cumulative distribution function or by the change-of-variables formula with a Jacobian factor. Forgetting the Jacobian is the standard error.