TWO DIMENSIONAL JOINT PDF

A random variable (X,Y) is of continuous type if [katex] F_{X,Y}(x,y) [/katex] is continuous, i.e. if there exists a non-negative function f(x,y) such that for every [katex] (x,y) \in \mathbb{R^2} [/katex],

F_{X,Y}(x,y) =  \int_{-\infty }^{x}  \int_{-\infty }^{y}  f(u,v) dv du

If [katex] F_{X,Y}(x,y) [/katex] has partial derivates of order upto two at (x,y) then-

\frac{\delta^2}{\delta x \delta y}  F_{X,Y}(x,y)  = f(x,y)

The function f(x,y) is called the joint pdf of (X,Y).

MARGINAL PDF

If (X,Y) is a continuous random variable with joint pdf f(x,y) then,

f_X(x) = \int_{-\infty}^{\infty} f(x,y) dy \\
f_Y(y) = \int_{-\infty}^{\infty} f(x,y) dx

are called the marginal pdf of X and Y respectively.

CONDITIONAL PDF

Let (X,Y) be a continuous random variable. The conditional pdf of X given Y=y is:

f_{X/Y}(x/y) = \frac{f(x,y)}{f_Y(y)}

provided [katex] f_Y(y)>0 [/katex]

Also, provided [katex] f_X(x)>0 [/katex] the conditional pdf of Y given X=x is :

f_{Y/X}(y/x) = \frac{f(x,y)}{f_X(x)}

CONDITIONAL DISTRIBUTION FUNCTION

Let (X,Y) be a continuous random variable. The conditional d.f. of X given Y is defined as:

F_{X/Y}(x/y) = \int_\infty^x f_{X/Y}(u/y).du

for all y for which [katex] f_Y(y) >0 [/katex].

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