Bivariate and Conditional Distributions

(HC chapter 2)

Rather than use $f,g,h,f_1,f_2$, etc as function names for pdf's, we will almost always use $f$, and if there is more than one random variable in the problem we will use a subscript to indicate the name of the variable whose pdf we are identifying.

For example, we may say that the pdf of $X$ is $f_X(x)=\alpha\,e^{- \alpha x}, \ x>0$. Of course, the $x$ could be replaced by any other letter. It is the $f_X$ that determines the function, not the $(\cdot )$. A similar notation is used for cumulative distribution functions. In the case of a conditional pdf, we will use, for example, $f_{X\vert Y=y}(x)$ for the conditional pdf of $X$ given $Y=y$. An alternative notation is $f(x\vert y)$.

Read HC 2.2 where most of the ideas should be familiar to you.

The two variables of a bivariate density $f_{X,Y}$ are correlated so the outcome due to one is influenced by the other. The conditional density $f_{Y\vert X}$ allows us to make statements about $Y$ if we have information on $X$. Recall that when we integrate out the terms in $X$ (or average over $f_X$), to get a density in $Y$ only (ie $f_Y$), we call that the marginal density of $Y$.

Definition 2..2
The conditional density function of $Y$ given $X=x$ is defined to be

$$f_{Y\vert X=x}(y)=\frac{f_{X,Y}(x,y)}{f_X(x)} \ \ \mbox{for} \ \ f_X(x)>0$$ (2.2)

and is undefined elsewhere.

Comments

1. In $f_{Y\vert X=x}(y)$, $x$ is fixed and should be thought of like a parameter.
2. $f_{X,Y}(x,y)$ is a surface above the $xy$-plane. A plane perpendicular to the $xy$-plane on the line $x=x_0$ will intersect the surface in the curve $f_{X,Y}(x_0,y)$. The area under this curve is then given by $\int^\infty_{-\infty}f_{X,Y}(x_0,y)\,dy =f_X(x_0)$. So dividing $f_{X,Y}(x_0,y)$ by $f_X(x_0)$ we obtain a pdf which is $f_{Y\vert X=x_0}(y)$.

Definition 2..3

If $X$ and $Y$ are jointly continuous, then the conditional distribution function of $Y$ given $X=x$ is defined as

$$F_{Y\vert X=x}(y) = P(Y\leq y\vert X=x)$$

$$= \int^y_{-\infty} f_{Y\vert X=x}(y)\,dy$$ (2.3)

for all $x$ such that $f_X(x)>0$.