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Distribution function

What is the probability that the realization of a random variable will be less than or equal to a certain threshold value? The distribution function of a random variable allows to answer exactly this question. Its value at a given point is equal to the probability of observing a realization of the random variable below that point or equal to that point.


The distribution function is also often called cumulative distribution function (abbreviated as cdf).


The following is a formal definition.

Definition If X is a random variable, its distribution function is a function [eq1] such that[eq2]where [eq3] is the probability that X is less than or equal to x.


Suppose a random variable can take only two values (0 and 1), each with probability 1/2. Its distribution function is[eq4]


Every distribution function enjoys the following four properties:

  1. Increasing. [eq5] is increasing, i.e., [eq6];

  2. Right-continuous. [eq5] is right-continuous, i.e.,[eq8]for any $x\in \U{211d} $;

  3. Limit at minus infinity. [eq5] satisfies [eq10]

  4. Limit at plus infinity. [eq5] satisfies [eq12]

Not only any distribution function enjoys these properties, but also, for any given function enjoying these four properties, it is possible to define a random variable that has the given function as its distribution function. The practical consequence of this fact is that when you need to check whether a given function is a proper distribution function, you just need to check that it satisfies the four properties above.

More details

More details about the distribution function can be found in the lecture entitled Random variables.

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