Joint distribution function

The joint distribution function (JDF) of a set of NN random variables defined in the same sample space is the function that describes the Probability of every possible combination of outcomes. The Probability distribution of all the variables together is called the joint distribution. For instance, if the variables are four dice being rolled together, the joint distribution function answers the question "what's the probability I'll roll 3, 4, 1 and 5?"

Formally, it is the function f(x1,…,xN)f(x_{1},\ldots,x_{N}) that, when integrated over all desired intervals I1∈Ω1,…,IN∈ΩNI_{1}\in \Omega_{1},\ldots,I_{N}\in \Omega_{N}, returns the probability of all variables being observed in their respective interval:

P(X1=x1,…,XN=xN)=∫IN…∫I1f(x1,…,xN) dx1…dxNP(X_{1}=x_{1},\ldots,X_{N}=x_{N})=\int_{I_{N}}\ldots \int_{I_{1}}f(x_{1},\ldots,x_{N})\ dx_{1}\ldots dx_{N}

For independent variables, the joint distribution function is just the product of each individual Probability density function or Probability mass function:

f(x1,,…,xN)=f1(x1)…fN(xN)f(x_{1},,\ldots,x_{N})=f_{1}(x_{1})\ldots f_{N}(x_{N})

For iid variables, it's simply the product of the shared distribution g(x)g(x)

f(x1,…,xN)=∏i=1Ng(xi)f(x_{1},\ldots,x_{N})=\prod_{i=1}^{N} g(x_{i})

Properties

  • It is normalized: ∫ΩN…∫Ω1f(x1,…,xN) dx1…dxN=1\int_{\Omega_{N}}\ldots \int_{\Omega_{1}}f(x_{1},\ldots,x_{N})\ dx_{1}\ldots dx_{N}=1 where the Ω\Omega are the sample spaces of the variables.