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

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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