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UD / ELEG / ELE 310 / How do we solve for characteristic functions?

How do we solve for characteristic functions?

How do we solve for characteristic functions?

Description

ELEG 310

Lectures 11 and 12


Given mean E[A]=m, variance VAR[A]=

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Y̲̅If you want to learn more check out How much blood passes through the glomeruli every minute?

                   this has no distribution, so its treated as a constant, If you want to learn more check out What are some signs of depression?

Only a changes;                                since time is not random

It is random

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                        ↘given

So

[Y̲̅]=

                using properties

VAR[Y̲̅] =        using theorems directly


New Topic: Characteristic Functions

The characteristic function of a random variable is given by

        1st interpretation: expected value of a function of ,

where w is unspecified

        2nd interpretation: the fourier transform of the

        

Inverse Fourier transform will be used:

                

Every pdf and its characteristic function form a unique Fourier transform pair.

Example 1: Exponential Random Variable

                        

                                        

                                        

                                        

Example 2: Uniform Variable - suppose is a uniform variable

        

                

                


Both Examples 1 and 2 are continuous-time examples.

What about discrete time?

 If discrete random variable takes on integer values only

 ↓                                                                                  

integer value                                        density                delta function

Locations where discrete

random variable have values

Characteristic Function:

        

        


Example 3: Bernoulli Random Variable, a.k.a “easiest possible case”

Fair coin         

                        

                        

                        

                        

Example 4: Using Characteristic Function

Derivative

        

                

        

                        using formula

                        

                                                =         ✔

        constant

                        

                                                

                                                         ✔

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