Key Points The probability of each value of a discrete random variable is between 0 and 1, and the sum of all the probabilities is equal to 1.

Random Variables

In probability and statistics, a randomvariable is a variable whose value is subject to variations due to chance (i.e. randomness, in a mathematical sense). As opposed to other mathematical variables, a random variable conceptually does not have a single, fixed value (even if unknown); rather, it can take on a set of possible different values, each with an associated probability.

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A random variable"s possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the possible outcomes of a past experiment whose already-existing value is uncertain (for example, as a result of incomplete information or imprecise measurements). They may also conceptually represent either the results of an "objectively" random process (such as rolling a die), or the "subjective" randomness that results from incomplete knowledge of a quantity.

Random variables can be classified as either discrete (that is, taking any of a specified list of exact values) or as continuous (taking any numerical value in an interval or collection of intervals). The mathematical function describing the possible values of a random variable and their associated probabilities is known as a probability distribution.

Discrete Random Variables

Discrete random variables can take on either a finite or at most a countably infinite set of discrete values (for example, the integers). Their probability distribution is given by a probability mass function which directly maps each value of the random variable to a probability. For example, the value of $x_1$ takes on the probability $p_1$, the value of $x_2$ takes on the probability $p_2$, and so on. The probabilities $p_i$ must satisfy two requirements: every probability $p_i$ is a number between 0 and 1, and the sum of all the probabilities is 1. ($p_1+p_2+\dots + p_k = 1$)


Discrete Probability Disrtibution

This shows the probability mass function of a discrete probability distribution. The probabilities of the singletons {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.

Examples of discrete random variables include the values obtained from rolling a die and the grades received on a test out of 100.

Continuous Random Variables

Continuous random variables, on the other hand, take on values that vary continuously within one or more real intervals, and have a cumulative distribution function (CDF) that is absolutely continuous. As a result, the random variable has an uncountable infinite number of possible values, all of which have probability 0, though ranges of such values can have nonzero probability. The resulting probability distribution of the random variable can be described by a probability density, where the probability is found by taking the area under the curve.


Probability Density Function

The image shows the probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important continuous random distribution. As notated on the figure, the probabilities of intervals of values corresponds to the area under the curve.

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Selecting random numbers between 0 and 1 are examples of continuous random variables because there are an infinite number of possibilities.