Week 06 notes

Continuous families: summary

01 Theory

Memorize this info!

Uniform:

• All times equally likely.

Exponential:

• Measures wait time until first arrival.

Erlang:

• Measures wait time until arrival.

Normal:

• Limiting distribution of large sums.

Normal distribution

02 Theory

Normal distribution

A variable has a normal distribution, written , when it has PDF given by:

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

The standard normal CDF is 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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• Check that
  • Observe that is an odd function, i.e. symmetric about the -axis.
  • One must also verify that the integral converges.
• Check that
  • Since , we find:
  • Use integration by parts to compute that . (Select and .)

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Generalized normal distributions are related to standard normal distributions by linear transformations. The generalized normal satisfies , where .

Let us check directly that by showing their PDFs are equal. Computing the CDF of , we find:

Then we can find the PDF of by differentiating :

Now since we know , we can infer that and .

03 Illustration

Example - Basic generalized normal calculation

Example - Gaussian: interval of

Example - Heights of American males

Example - Variance of normal from CDF table