- 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
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.
Example 6.3 on page 187.
- 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
- 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.
- 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.