1. Mean & variance of random variables
Chapter 7

2. Expected Value The mean (center) of a discrete random variable, X.
It is a weighted average
-Each value of X is weighted by its probability.

3. Finding Expected Value
To Find:
Or in other words:
Multiply each value of X by its probability and add products
This terminology is often used in relation to game of chance or business prediction.

4. Using Expected Value Example:
Matthew is a doorman. The following table gives the probabilities that customers will give various amounts of money:
What size tip can Matthew expect to get?
Amounts of $ 30 35 40 45 50 55 60
Probability
.45
.25
.12
.08
.05
.03
.02

5. Example 2
Concessionaires know that attendance at a football stadium will be 60,000 on a clear day, 45,000 if there is light snow and 1500 if there is heavy snow. Furthermore the probabilities of clear skies, light snow or heavy snow on a particular day is 1/2, 1/3, and 1/6 respectively. What average attendance should be expected for the game?

6. Example 3
10,000 raffle tickets are sold at $2 each for four prizes of $500, $1000, $1500, and $ If you buy one ticket what is the expected value of your gain?

7. Law of large numbers
As the number of observations increases, the mean of the observed values, , approaches the mean of the population, µ.
The more variation in the outcomes, the more trials are needed to ensure that is close to µ .

8. Rules for means
If X is a random variable and a and b are fixed numbers, then
If X and Y are random variables, then
We use these for conversions!!!

9. example
The equation Y = X converts a PSAT math score, X, into an SAT math score, Y. The average PSAT math score is What is the average SAT math score?

10. example
Let represent the average SAT math score. Let represent the average SAT verbal score. What is the average combined score?

11. Variance of a discrete random variable
If X is a discrete random variable with mean , then the variance of X is
The standard deviation is the square root of the variance.

12. Rules for variance
If X is a random variable and a and b are fixed numbers, then
If X and Y are independent random variables, then

13. example
The equation Y = X converts a PSAT math score, X, into an SAT math score, Y. The standard deviation for the PSAT math score is 1.5 points. What is the standard deviation for the SAT math score?

14. example
The standard deviation for the SAT math score is 150 points, and the standard deviation for the SAT verbal score is 165 points. What is the standard deviation for the combined SAT score?
NOT INDEPENDENT VARIABLES, THE RULE FOR ADDING VARIANCES DOES NOT APPLY!!!

15. Example
X and Y are independent random variables. If and find

16. One More…
As head of inventory for Knowway computer company, you were thrilled that you had managed to ship 2 computers to your biggest client the day the order arrived. You are horrified, though, to find out that someone restocked refurbished computers in with the new computers in your storeroom. The shipped computers were selected randomly from 15 computers is stock, but 4 of those were actually refurbished.
If your client gets 2 new computers, things are fine. If the client gets a refurbished computer, it will be sent back at your expense - $100- and you can replace it. However, if both computers are refurbished, the client will cancel the order for this month and you will lose $ What is the expected value and the standard deviation of your loss?