# Conditional Probability Density Functions

[Note: This post was created as part of a lecture for STAT 131 at UCSC.]

Recall that for two continuous random variables $$X$$ and $$Y$$, we work with the joint probability density function $$f(x,y)$$. Below, we plot $$f(x,y)$$ for a pair of random variables.

In some cases, we might be interested in probabilities for one random variable, conditioning on the other. For example, for $$X$$ and $$Y$$ in the previous plot, we might want to know how the probabilities for $$X$$ change given that we know $$Y=1$$. Our intuition might tell us to look at the joint pdf with $$y=1$$ in this case (i.e.Â $$f(x,1)$$). This is plotted below as the orange curve. Notice that the vertical slice between our $$x-y$$ plane and the orange curve looks very similar to a probability density for a single variable.

The primary issue here is that this region does not integrate to one, and thus does not constitute a valid probability density. However, we can scale this region so that it does integrate to one. It turns out that the scaling constant $$c$$ in this case is $$c=1/f_Y(1)$$. Recall that $$f_Y(1)$$ is the marginal density of $$Y$$ evaluated at $$y=1$$. After rescaling, we plot the new curve in green. Note that this is a valid probability density. In fact we call this the conditional probability density of $$X$$ given that $$Y=1$$, or $$f_{X|Y=1}(x|1)$$.

This example shows that we obtained a valid conditional density by rescaling the joint density at the set of points where $$y=1$$, $f_{X|Y=1}(x|1) = \frac{f(x,1)}{f_Y(1)}.$ We could repeat this process conditioning on any other value of $$Y=y$$, resuling in the general formula $f_{X|Y}(x|y) = \frac{f(x,y)}{f_Y(y)}.$

##### Paul A. Parker
###### Assistant Professor

My research interests include Bayesian methods, especially when applied to dependent data scenarios, often using survey data.