Marginal distribution function

The marginal distribution function (MDF) of a set of $N$ random variables is the function that gives the Probability of a specific outcome for one of the variables, regardless of what the other variables do. For instance, if the variables are four dice being rolled together, the marginal distribution function answers the question "what's the probability that the first die will roll a 3? I don't care what the other three dice do."

Formally, given a Joint distribution function $f(x_{1},\ldots,x_{N})$ , the marginal distribution function is

f_{M}(x_{1})=\int_{\Omega_{N}}\ldots \int_{\Omega_{2}}f(x_{1},\ldots,x_{N})\ dx_{2}\ldots dx_{N}

where the $\Omega$ are the sample spaces of the random variables. In other words, you are integrating over the sample spaces of all variables except the one that matters to you. This "gets rid" of all other variables by considering all the possible cases they can take and leaves you with just the one you are interested in.

Properties#

It is normalized: $\int_{\Omega}f_{M}(x)=1$ .
If the variable is independent of all others, its MDF matches its Probability density function. In fact, for independent variables the JDF is simply the product of PDFs

so the nested integrals come out and become a simple product $$f_{M}(x_{1})=f_{1}(x_{1})\int_{\Omega_{2}}f_{2}(x_{2})\ dx_{2}\ldots \int_{\Omega_{N}}f_{N}(x_{N})\ dx_{N}

but by definition, the integral of a PDF over its entire sample space is 1, so all the integrals evaluate to 1 and we're left with

The MDF differs from the PDF of the random variable only if it's correlated to other variables. Intuitively, this makes sense: the marginal distribution, in a way, measures dependence. If two variables are independent, then the outcome of one won't affect the other, so the marginal distribution won't be affected. It is related to the conditional distribution function.