Continuous random variable

Normal distribution

The normal distribution or bell-curve is occurs frequently over a diverse range of applications.

It has the following probability density function

with no closed-form Cumulative distribution function. One important property of the normal distribution is that it is symmetric about .

Graph

Properties

Let .

  1. Expectation:
  2. Variance:
  3. Moment-generating function:

Furthermore

  1. If and are independently distributed then .
  2. By Cramér's theorem the converse of the above also holds.
Proof of 1

By ^P1

as required.

Standard form

See Standard normal distribution.

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