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
✔