Understanding the marginal probability from joint pdf

Question

What is the intuitive meaning for the marginal density probability of a joint pdf? I mean, graphically, what does it represent? Lets say I have the pdf $f_{X, Y}(x, y)$ for $0 < x < 1, 0 < y < x$. Drawing the domain I have the area under $y = x$. The pdf is "above" this domain. So far so good.

But in order to calculate the marginal pdf I'll have $f_X(x) = \int_{-\infty}^{+\infty}f_{X, Y}(x, y)dy$ - why? What is the reasoning behind this? This is surely not the "projection" of $f_{X, Y}$ in the $zOx$ plane; then what is it exactly?

And what if it where $f_{X|Y}$?

Answer 1

A mental picture of what is going on with the marginal pdf is imagining telescoping the joint pdf from two to a single dimension, i.e. integrating one of the variables out for each point in the domain of the variable whose marginal you are interested in.

Here is a graphical picture of the following joint pdf:

$$f_{X,Y}(x,y) = 4x $$

where

$$\begin{cases}0 \leq x \leq 1 \\ 0 \leq y \leq x^2 \end{cases}$$

To get the marginal $f_Y(y)$ the random variable $X$ has to be integrated out:

$$f_Y(y)=\int_{x=\sqrt y}^{x=1}4x \, dx = 2(1-y)$$

Answer 2

The marginals have a nice geometric interpretation that dovetails with the probabilistic interpretation. For an example, consider two independent dice rolls $X$ and $Y$, both valued in $\{1,2,3,4,5,6\}$, of course. The joint probability density $f_{X,Y}(x,y)$ is then $1/36$ for each $x,y$. (I recommend drawing a table with an $X$-axis and a $Y$-axis with the probabilities of each outcome in each cell of the table.) To find the probability $f_X(x)$, that is, that you roll an $x$ with $X$, you can sum over all the possibilities for your roll with $Y$: $$ f_X(x) = \sum_{y=1}^6f_{X,Y}(x,y). $$ Intuitively, what this means is the total probability is spread over all the outcomes for $X$ and $Y$, so to find $P(X=x)$, you may integrate or sum $P(X=x,Y=y)$ over the space of outcomes $y$ for $Y$. When you consider conditional probability, for example, you want to know $P(X=x|Y=y)$, you simply restrict your total space of outcomes to the row where $Y=y$. You can build your intuition for both conditional and marginal laws with this with simpler discrete example.

For a first brush with the subject, it isn't bad to think of the continuous case where $f_{X,Y}$ is a function on the plane as a limit of this special case when $X$ and $Y$ are dice rolls with many sides (and possibly different laws other than the uniform law), and you draw finer and larger tables to accommodate the increasing number of outcomes.