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:
- See also Standardizing a Random Variable
rmk. Empirical Rule: Rule of thumb for calculating probabilities (integrals) of normal distributions
- ! Generalized version: Chebyshev’s Inequality
- 1 std. dev. away is ; 2 std.dev. away is
Estimators
let
- ⇒ Log likelihood:
Score
One R.V. | Multiple Data |
---|---|
MLEs
Fisher Information
- Unknown , known
- Known unknown