Characteristic functions - Probability and probability Distributions, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

There are random variables for which the moment generating function does not exist on any real interval with positive length. For example, consider the random variable X that has a Cauchydistribution

You can show that for any nonzero real number s

Therefore, the moment generating function does not exist for this random variable on any real interval with positive length. If a random variable does not have a well-defined MGF, we can use the characteristic function defined as

where  and ω is a real number. It is worth noting that e^jωX is a complex-valued random variable. We have not discussed complex-valued random variables. Nevertheless, you can imagine that a complex random variable can be written as X=Y+jZ, where Y and Z are ordinary real-valued random variables. Thus, working with a complex random variable is like working with two real-valued random variables. The advantage of the characteristic function is that it is defined for all real-valued random variables. Specifically, if X is a real-valued random variable, we can write

Therefore, we conclude

The characteristic function has similar properties to the MGF. For example, if X and Y are independent

More generally, if X₁,X₂, ..., X_n are nn independent random variables, then