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

                                                         ✔