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 E1 ∪ E2 is an event containing all posible outcomes in either E1 or E2.

The intersection of two events, denoted E1 ∩ E2 is an event containing all posible outcomes in both E1 and E2.

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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