Transform methods

As pointed out in the units on Expectation and Variance , the mathematical expectation $E\left[X\right]={\mu }_X$ of a random variable X locates the center of mass for the induced distribution, and the expectation

measures the spread of the distribution about its center of mass. These quantities are also known, respectively, as the mean (moment) of X and the second moment of X about the mean. Other moments give added information. For example, the third moment about the mean $E\left[{(X-{\mu }_X)}^3\right]$ gives information about the skew, or asymetry, of the distribution about the mean. We investigatefurther along these lines by examining the expectation of certain functions of X . Each of these functions involves a parameter, in a manner that completely determines the distribution.For reasons noted below, we refer to these as transforms . We consider three of the most useful of these.

Three basic transforms

We define each of three transforms, determine some key properties, and use them to study various probability distributions associated with random variables. In the section on integral transforms , we show their relationship to well known integral transforms. These have been studied extensivelyand used in many other applications, which makes it possible to utilize the considerable literature on these transforms.

Definition . The moment generating function M _X for random variable X (i.e., for its distribution) is the function

The characteristic function φ _X for random variable X is

The generating function $g_X\left(s\right)$ for a nonnegative, integer-valued random variable X is

The generating function $E\left[s^X\right]$ has meaning for more general random variables, but its usefulness is greatest for nonnegative, integer-valued variables, and we limit ourconsideration to that case.

The defining expressions display similarities which show useful relationships. We note two which are particularly useful.

Because of the latter relationship, we ordinarily use the moment generating function insteadof the characteristic function to avoid writing the complex unit i . When desirable, we convert easily by the change of variable.