*Random variable*: A variable that assumes values in a irregular pattern (or no particular pattern).*Discrete random variables*: Let*X*be a random variable. If the number of possible values of*X*is finite, or countably infinite,*X*is called a discrete random variable.Let be all possible values of

*X*, and be the probability that , then must meet the following conditions.- for all .

Examples: example 6.1 and 6.2 on p. 186

*Contineous random variables*: If the values of*X*is an interval or a collection of intervals, then*X*is called a contineous random variable.For contineous random variable, its probability is represented as

The function is called the probability density function (pdf) of the random variable , which has to meet the following condition.- for all .
- .
- if is not in .

*Cumulative distribution function.*The cumulative distribution function (cdf), denoted by , measures the probability that the random variable assumes a value less than or equal to , .- If is discrete, then
- If is contineous, the

Some propertities of cdf include:

- is a non-decreasing function. If then .
- .
- .

Example: 6.4, 6.5 on page 189.

*Expectation and variance.*Expectation essentially is the expected value of a random variable. Variance is a measure how a random variable varies from its expected value.- For discrete random variables

- For continueous random variables

- For discrete or continueous random variables, its variance is

which has an identity

- A more frequently used practical measure is
*standard deviation*of a random variable, which is expressed as the same units as that of expectation.

Examples: 6.6, 6.7 on page 191.

- For discrete random variables
*The mode.*The mode is used to describe most frequently occured values in discreate random variable, or the maximum value of a continueous random variable.