Some distribution functions:
where is a cumulative distribution function.
Hazard rate function
where is a probability density function.
Cumulative hazard rate function
The following relationship is often useful:
Or more generally,
When , the expected value of such a function is called the nth raw moment and is denoted by . Let be the first raw moment. That is, . is called an nth central moment.
Moments are used to generate some statistical measures.
The coefficient of variation is .
Covariance of two distribution functions
*Note: if and are independent,
All of the above definitions should be memorized. Some things that might be tested in the exam are:
- Given a particular distribution function, what happens to skewness or kurtosis in the limit of a certain parameter?
- What is the expected value, variance, skewness, kurtosis of a given distribution function?
- What is the covariance or correlation coefficient of two distribution functions?
Central moments can be calculated using raw moments. Know how to calculate raw moments using the statistics function on the calculator. This can be a useful timesaver in the exam. Using alternating positive and negative binomial coefficients, write an expression for with and as the two binomial terms.
Since , the two terms on the end simplify to . The result is
Moment Generating Function:
If the moment generating function is known for random variable , it’s nth raw moment can be found by taking the nth derivative of and evaluating at 0. Moment generating functions take the form:
If , then .