TABLE OF CONTENTS The concept of joint probability is a very important one in signal analysis. Very often the values of two variables from the same or different sets of measurements are being compared or studied and a two-dimensional sample space exists. The joint or bivariate probability density function and its moments are the basis for describing any interrelationships or dependencies between the two variables. A simple example is the selection of a resistor from a box of resistors. The random variables are the resistance, r, and wattage, w. If a resistor is selected, it is desired to know the probabilities associated with ranges of resistance and wattage values, that is
| (4.19) |  |
where R and W are particular values of resistance and wattage, respectively. For signals this concept is extended to describe the relationship between values of a process x( t) at two different times, t 1 and t 2, and between values of two continuous processes, x( t) and y( t), at different times. These probabilities are written as
| (4.20) |  |
respectively. The joint probabilities are functionally described by bivariate probability distribution and density functions. The bivariate cdf is
| (4.21) |  |
It is related to the bivariate pdf through the double integration
| (4.22) |  |
These functions have some important properties. These are
| (4.23) |  |
Again, when there is no confusion concerning the random variable and its specific values, a simpler notation is often utilized.
