Marginal and conditional distributions, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

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Joint and Marginal Distributions: Suppose the random variables X and Y have joint probability density function (pdf) f_X,Y(x,y). The value of the cumulative distribution function F_Y(y) of Y at c is then 
  F_Y(c) = P( Y ≤ c)   = P(-∞ < X < ∞, Y ≤ c)   = the volume under the graph of f_X,Y(x,y) above the region ("half plane") 

Setting up the integral to give this area, we get 

Thus the pdf of Y is f_Y(y) = F_Y'(y) = g(y). 
 In other words, the marginal pdf of Y is  

Similarly, the marginal pdf of X is 

In words: The marginal pdf of X is ___________________________________________ 
 
 Note: When X or Y is discrete, the corresponding integral becomes a sum

Joint and Conditional Distributions: 

First consider the case when X and Y are both discrete. Then the marginal pdf's (or pmf's = probability mass functions, if you prefer this terminology for discrete random variables) are defined by    

f_Y(y) = P(Y = y) and  f_X(x) = P(X = x). 

 The joint pdf is, similarly,  

f_X,Y(x,y) = P(X = x and Y = y).  

The conditional pdf of the conditional distribution Y|X is 

In words:  
 
Is this also true for continuous X and Y? In other words: 

It is enough to show that . (Draw a picture to help see why!).  

Starting with the right side, we can reason as follows:  

(Draw pictures to help see the steps!) 

Summarizing: The conditional distribution Y|X has pdf 

In word equations: 

Conditional density of Y given

(and, of course, the symmetric equation holds for the conditional distribution of X given Y).