Theory

Theory 1

Normal distribution

A variable has a normal distribution, written or “ is Gaussian ,” when it has PDF given by:

The standard normal is and its PDF is usually denoted by :

The standard normal CDF is usually denoted by :

• To show that is a valid probability density, we must show that .
  • This calculation is not trivial; it requires a double integral in polar coordinates!
• There is no explicit antiderivative of
  • A computer is needed for numerical calculations.
  • A chart of approximate values of is provided for exams.

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• To check that :
  • Observe that is an odd function, i.e. symmetric about the -axis.
  • One must then simply verify that the improper integral converges.
• To check that :
  • Since , we find:

• Use integration by parts to compute that . (Select and .)

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General and standard normals

Assume that and are constants. Define . Then:

That is, has the distribution type .

Derivation of PDF of

Suppose that . Then:

Differentiate to find :

From this fact we can infer that and whenever .