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UD - ELE 310 - Class Notes - Week 7

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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

✔