def. A distribution gives comprehensive information about an experiment. A distribution can be a table showing the probabilities of all outcomes, or a probability density function.

def. let Ω be an outcome space. All functions satisfying the following criteria are probability distributions.


  1. and
  2. For all ,
  3. If A and B are disjoint then

The 3rd condition can also be generalized for any distributions:

  • If are pairwise disjoint, then .

Remark. The distribution for countable, discrete outcome spaces follow the above axiom.

For a random variable where h is the height of some population, the probability that is the shaded area:


Calculated by:

Probability Mass Function

def. Probability Mass Function. For a discrete random variable , the probability mass function is the function that gies the probability of all values of .

thm. Addition Rule for Random Variables. For a discrete random variable the following is true:

Probability Density Function

def. Probility Density Function. is a probility density function of random variable iff:

def. Cumilitave Density Function. is the cumulative density function of random variable :

if and only if:

  1. is never decreasing over its domain


Relationship between and is a derivate and antiderivative.

Note that when you get you will get a integration constant . You can get rid of this by using the property .

List of Common Distributions