Random variable

A random variable is a measurable function from a probability space to a value space. It represents outcomes as values whose probabilities can be studied.

Probability space: A mathematical structure consisting of outcomes, a collection of events, and a probability measure. It supplies the domain and probability measure that make a random variable precise.
Probability distribution: A probability measure describing how probability is allocated across possible values. It summarizes the probabilities produced when the random variable maps outcomes to values.
Bernoulli distribution: A distribution on zero and one, determined by the probability of one. A single success-or-failure trial is modeled by a Bernoulli random variable.
Discrete random variable: A random variable whose possible values form a finite or countable set. Its individual values can carry positive probability masses.
Measurable function: A function whose inverse images of measurable sets are measurable. This condition ensures that events defined by random-variable values have probabilities.
Cumulative distribution function: The function giving the probability that a random variable is at most each specified value. It records the random variable's distribution through threshold probabilities.
Binomial distribution: The distribution of the number of successes in a fixed number of independent Bernoulli trials. It models a random variable counting successes across repeated trials.
Continuous random variable: A random variable whose distribution is continuous, commonly described by a density. Unlike a discrete variable, it typically assigns zero probability to each individual value.
Sample space: The set of all possible outcomes in a probabilistic model. Each outcome in this set is mapped to a value by the random variable.
Probability mass function: A function giving each possible value of a discrete random variable its probability. It describes value probabilities when the random variable has a discrete range.

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