202405310836
Status: #idea
Tags: Probability Distributions
State: #nascient

Normal Distribution

The Normal distribution is the undisputed queen of Statistics, being the distribution that is the most ubiquitous. It is used irrespective of whether it fits, because of its nice properties.

PDF

For a standard normal, the pdf is:

f (z) = \frac{1}{\sqrt{2 π}} e x p (- \frac{z^{2}}{2})

If we have a random variable $X \sim N (μ, σ^{2})$ , then its pdf will be:

f (x) = \frac{1}{2 π σ^{2}} e x p (- \frac{(x - μ)^{2}}{2 σ^{2}})

Moment Generating Function for STANDARD Normal

So
If the $t^{m}$ does not occur in the Taylor Series Expansions of your MGF, then you can infer that the moment in question is 0.

MGF for GENERAL Normal

Properties

Given $X_{i} \sim N (μ_{i}, σ_{i}^{2})$ with $i = 1, 2, \dots, n$ where the $X_{i}$ are independent, we want $Y = \sum_{i = 1}^{n} a_{i} X_{i}$ :
Then the MGF for that linear combination will be

M G F_{Y} = e x p (t (\sum_{i = 1}^{n} a_{i} μ_{i}) + \frac{t^{2}}{2} (\sum_{i = 1}^{n} a_{i}^{2} σ_{i}^{2}))

Such that the mean and variance are just:

$μ_{Y} = \sum_{i = 1}^{n} a_{i} μ_{i}$
$σ_{Y}^{2} = \sum_{i = 1}^{n} a_{i}^{2} σ_{i}^{2}$

By the above we get this supremely important result

Normal Distribution

PDF

Moment Generating Function for STANDARD Normal

MGF for GENERAL Normal

Properties

Relevant Links