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

Normal Distribution

PDF

MGF 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 $X_i\sim N({\mu }_i,{\sigma }_i^2)$ 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

Such that the mean and variance are just:

• ${\mu }_Y=\sum _{i=1}^n{a}_i{\mu }_i$
• ${\sigma }_Y^2=\sum _{i=1}^n{a}_i^2{\sigma }_i^2$

By the above we get this supremely important result

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