Expectation and other distribution parameters

1. Section 05 Expectation and other distribution parameters

2. Expected value • The expected value, or expectation, is the average or mean of the random variable • Denoted • Discrete: • Continuous:

3. Moments of a random variable • If n is a positive integer… • The n-th moment of X is • The n-th central moment of X (about the mean) is

4. Variance • The variance of X is denoted , , , or • Of course, the standard deviation is the square root of the variance, so

5. Moment Generating Function (MGF) • Discrete: • Continuous: • Properties:

6. Other distribution characteristics • Percentile • A value of X denoted such that of the distribution of X is less than • The median is where p=.5 • Mode • Point where the pmf or pdf is maximized • Skewness= • Positive skewness – distribution skewed right • Negative skewness – distribution skewed left

7. Expectation and Variance of Functions • Expectation: Constants out, coefficients out • Variance: Constants disappear, coefficients out as squares

8. Sample Exam #60 A recent study indicates that the annual cost of maintaining and repairing a car in a town in Ontario averages 200 with a variance of 260. If a tax of 20% is introduced on all items associated with the maintenance and repair of cars (i.e., everything is made 20% more expensive), what will be the variance of the annual cost of maintaining and repairing a car?

9. Sample Exam #44 An insurance policy pays 100 per day for up to 3 days of hospitalization and 50 per day for each day of hospitalization thereafter. The number of days of hospitalization X is a discrete random variable with probability function Determine the expected payment for hospitalization under this policy.

10. Sample Exam #62 A random variable X has the cumulative distribution function Calculate the variance of X