-
Experimental focus
- Many observable characteristics in experiments
- Experimenters typically focus on specific aspects of a sample
-
Examples:
- Commuting patterns study
- Focused characteristics:
- Commuting distance
- Number of people in the same vehicle
- Ignored characteristics:
- IQ
- Income
- Family size
- Other unrelated characteristics
- Focused characteristics:
- Component failure study
- Focused characteristic:
- Number of failures within 1000 hours
- Ignored characteristic:
- Individual failure times
- Focused characteristic:
- Commuting patterns study
-
Random Variable
- Definition: A rule of association that assigns a number to each outcome of an experiment
- Characteristics:
- Variable: Different numerical values are possible
- Random: The observed value depends on which experimental outcome occurs - Examples:
- Number of components that fail to last 1000 hours in a sample of ten
- Total weight of baggage for a sample of 25 airline passengers - Visual representation: Figure 3.1
Figure 3.1 A random variable
random variable
For a given sample space of some experiment, a random variable (rv) is any rule that
- associates a number with each outcome in .
In mathematical language, a random variable is a function
- whose domain is the sample space
- whose range is the set of real numbers.
- Notation for Random Variables
- Random Variables:
- Denoted by uppercase letters (e.g., X, Y)
- Usually chosen from near the end of the alphabet
- Specific Values of Random Variables:
- Denoted by lowercase letters
- Contrast with previous use of lowercase letters for variables
- Formal Notation:
- X(ω) = x
- X: The random variable
- ω: A specific outcome
- x: The value associated with outcome ω by random variable X
- X(ω) = x
- Random Variables:
- Specifying an rv X
- In Example 3.1, was specified by:
- Explicitly listing each element of
- The associated number
- However, this approach has a limitation:
- It can be tedious if S contains more than a few outcomes
- Fortunately, there is an alternative:
- Such a listing can frequently be avoided
- In Example 3.1, was specified by:
- In Examples 3.1 and 3.2, the only possible values of the random variable were 0 and 1.
- Such a random variable arises frequently and is given a special name.
- Named after the individual who first studied it.
Definition
Any random variable whose only possible values are 0 and 1 is called a Bernoulli random variable.
We will sometimes want to consider several different random variables from the same sample space.
EX 3.3 pumps of gas stations again
Each of the random variables of Examples 3.1-3.3 can assume only a finite number of possible values. This need not be the case.