Discrete Joint Distribution

def. Joint Distributions of two discrete random variables encode the probabilities for every pair of for . Following is an example of a joint distribution where is the result of rolling a first dice and the result of a second roll.

REMARK. Joint Distributions are distributions too, which means it has to follow all the rules of distributions (e.g. )

Continuous Joint Distribution

def. Joint Probability Density. Let be two independent continous random variables. Then the joint probability density function is defined as the derivative of the cumulative density function:

And thus the following holds:

  • where is an event

The blue volume in the picture is the probability of the event X and Y are in R.

The blue volume in the picture is the probability of the event X and Y are in R.

Full Visual Example


  • Blue is the probability density function,
  • Red is the marginal probability density of ,

Minimum and Maximum Joint Dist

thm. Let be i.i.d.; let . Then:

Examples of Joint Distributions


Recall that joint distributions are also distributions [=encapsulate fully the information of an experiement].

Uniform Joint

thm. If are both uniformly distributed over , then…

  • height of the distribution is (where denotes the area of the outcome space.)


Normal Joint (Linear Combination)

thm. Linear Combination of Normal Distributions. If and then:

Normal Joint (Product)

thm. if and (i.e. std. dev. is the same) then

  • Volume of sector from is


Rayleigh Distribution

[=Squared & Rooted Joint Normal]

def. Rayleigh Distribution. let and , then:

  • Where is the “scaling factor” (standard dev. must be same for )
  • If then is a Standard Rayleigh distribution:

thm. Standardizing Rayleigh Distributions. If :