**Some distribution functions:**

Survival function

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:

**Expected Value:**

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.

*Variance*

The *coefficient of variation* is .

*Skewness*

*Kurtosis*

*Covariance* of two distribution functions

*Note: if and are independent,

*Correlation coefficient *

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?

## Additional Notes

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.

Example:

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 .