def. Normal Distribution. A random variable distributed over a Normal distribution with mean and standard deviation is denoted:

def. Cumulative Distribution Function (CDF) of a Normal Distribution

Observe:

def. Standard Normal Distribution. A standard normal distribution is a normal distribution where

Tip

You can approximate a bunch of distributions using the Normal.

rmk. Linear Transformation of Normal Distribution. If , then

    • ⇒ Thus

remark. Exponentiating Transformation of Normal Distribution. If

(using Law of Unconscious Statistician)

rmk. Standardizing the Normal Distribution. Given :

  • has the standard normal distribution
  • The pdf is as follows:

rmk. Empirical Rule: Rule of thumb for calculating probabilities (integrals) of normal distributions

Estimators

let

  • ⇒ Log likelihood:

Score

One R.V.Multiple Data

MLEs

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