Characteristic funtion of RV

[cappadonza]

so a charateristic function of a RV is complex valued funtion. from my lecture, the distribution funtion of a Random variable is not always "well behaved", may not have a density etc. A charateristic function on the other had is "well behave".
What i don't understand is, is that the only reason we use it ?
how is it actually derived, why does it have to be complex valued .
this is the definition I'm given $$\phi(t) = \mathbb{E}(e^{itX})$$
how is this actually derived, is somewhere where i can find the proof ?

 

[mathman]

You need to clarify your question. Definitions aren't derived.

As far as usage, the simplest example is deriving the distribution function a sum of independent random variables. The characteristic function of the sum is the product of the characteristic functions of the individual variables.

 

[statdad]

if you are studying characteristic functions you should have already seen moment-generating functions.

moment generating functions can be used to uniquely identify the form of a distribution IF (big if) the moments of the distribution satisfy a very strict requirement. that doesn't happen all the time.

even worse, not every probability distribution has a moment-generating function: think of a t-distribution with 5 degrees of freedom: no moments of order 4 or greater, so no moment generating function.

however, EVERY distribution has a characteristic function, and every distribution is uniquely determined by the form of that function. that is one (not the only) reason for their importance.

 

[SW VandeCarr]

I found this presentation to be helpful in understanding the derivation and applications of the characteristic function.

http://www.sjsu.edu/faculty/watkins/charact.htm

 

[blue_raver22]

I can explain the concept of characteristic function of a random variable (RV) and its importance in probability theory.

A characteristic function is a complex-valued function that uniquely defines the distribution of a RV. It is defined as the expected value of the complex exponential function raised to the power of the RV. This definition may seem arbitrary, but it has important properties that make it useful in probability theory.

One of the main reasons we use characteristic functions is because they have nice mathematical properties that make them easier to work with compared to other distribution functions. For example, the characteristic function of a sum of independent RVs is equal to the product of their individual characteristic functions. This property is known as the convolution property and it simplifies the analysis of complex systems involving multiple RVs.

The characteristic function is also "well-behaved" in the sense that it always exists and is continuous, even for distributions that do not have a density function. This makes it a powerful tool for analyzing distributions that may not have a simple analytical form.

The reason why the characteristic function is complex-valued is because it involves the complex exponential function. This may seem daunting, but it has important advantages. Firstly, the complex exponential function has nice properties that make it easier to manipulate in mathematical calculations. Secondly, the real and imaginary parts of the characteristic function have important interpretations in terms of moments and cumulants of the distribution, respectively.

The derivation of the characteristic function relies on advanced mathematical techniques such as Fourier transforms and complex analysis. The proof of its properties can be found in advanced probability theory textbooks or online resources. However, as a scientist, it is important to understand the concept and its applications rather than the technical details of its derivation.

In conclusion, the characteristic function of a RV is an important tool in probability theory due to its nice properties and its ability to capture the distribution of a RV. Its complex-valued nature and derivation may seem complex, but it has significant advantages in analyzing and understanding complex systems.