Probability
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Continuous families: summary

01 Theory

Theory 2

Uniform:

  • All times equally likely.

Exponential:

  • Measures wait time until first arrival.

Erlang:

  • Measures wait time until arrival.

Normal:

  • Limiting distribution of large sums.
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Normal distribution

02 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.
  • 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 .)

General and standard normals

Assume that and are constants. Define . Then:

That is, has the distribution type .

From this fact we can infer that and whenever .

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03 Illustration

Example - Basic generalized normal calculation

Basic generalized normal calculation

Suppose . Find .

Solution

First write as a linear transformation of :

Then:

Look in a table to find that and therefore:

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Example - Gaussian: interval of

Gaussian: interval of 2/3

Find the number such that .

Solution

First convert the question:

Solve for so that this value is :

Use a table to conclude .

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Example - Heights of American males

Heights of American males

Suppose that the height of an American male in inches follows the normal distribution .

(a) What percent of American males are over 6 feet, 2 inches tall?

(b) What percent of those over 6 feet tall are also over 6 feet, 5 inches tall?

Solution

(a) Let be a random variable measuring the height of American males in inches, so . Thus , and:

(b) We seek as the answer. Compute as follows:

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Example - Variance of normal from CDF table

Variance of normal from CDF table

Suppose , and suppose you know .

Find the approximate value of using a table.

Solution

So and thus . Then:

so .

Looking in the chart of for the nearest inverse of 0.8, we obtain , hence .

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