This is a continuation of our previous post on Random Variables. If you missed it, go back and check it out.
A joint density function such as described above can be used to calculate probabilities about X and Y by summing f(x,y) over the appropriate values of x and y just as in the one-dimensional case.
It is important to note that the new sample space consists of points in the (x,y) plane, and the probabilities are given by f(x,y): f(x,y) > 0.
Joint Continuous Density Functions
As for discrete random variables, a density function for two or more continuous random variables is a generalization of a density function for one variable.
Thus, a density function for the continuous random variables X and Y is denoted by f(x,y).
Marginal and Conditional Distributions
Consider an experiment in which A is the event that a random variable assumes the values x and B is another event in which the random variable Y assumes the value y. By the application of the multiplication rule; i.e
P(A∩B) = P(A) P(B|A). It is easy to see that f(x,y) = f(x) f(y|x).
Since f(y|x) is the conditional probability that Y assumes the value y given that X has a known fixed value x. then the sum of f (y|x) with all possible values of y is equal to x. then the sum of f(y|x) gives possible values of y is equal to 1.
∑ f(x,y) = ∑f(x) f(y|x)
This result can be extended conveniently to the marginal distribution of continuous random variables.