# Probabilty

## Sample Spaces and Events

The set of all possible outcomes of a random experiment is called the *sample space* of the experiment. It is often denoted as **S**.

For example, if the experiment consists of rolling a six-sided die, then **S = {1,2,3,4,5,6}**. If the experiment consists of flipping a coin, then **S = {“Heads”, “Tails”}**.

An *event* is a subset of the sample space of a random experiment.

The *union* of two events, denoted **E _{1} ∪ E_{2}** is an event containing all posible outcomes in either

**E**or

_{1}**E**.

_{2}The *intersection* of two events, denoted **E _{1} ∩ E_{2}** is an event containing all posible outcomes in both

**E**and

_{1}**E**.

_{2}The *compliment* of an event, denoted **E’**, is the set of all outcomes in **S** that are not contained in **E**.

The following identities are often useful:

**(A ∪ B)’ = A’ ∩ B’**

**(A ∩ B)’ = A’ ∪ B’**

## Random Variables

A sample space is *discrete* if it consists of a finite (or countably infinite) set of outcomes.

A *random variable* is a function that assigns a real number to each outcome in the sample space of a random experiment..

A *discrete random variable* is a random variable with a finite (or countably infinite) range. Examples: number of defective parts out of 1000 tested; number of customer complaints in a week.

A *continuous random variable* is a random variable with an interval (either finite or infinite) for its range. Examples: length, pressure, voltage, weight, temperature.

## Sources

Montgomery, D. & Runger, G. (1999). *Applied Statistics and Probabilty for Engineers* [2nd Ed.]. New York: Wiley.

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