1. Special Topics

2. Mean of a Probability Model The mean of a set of observations is the ordinary average. The mean of a probability model is also an average, but with an essential change, not all outcomes are equally likely. It is actually a weighted average.

3. Formula for Mean of a Discrete Probability Model If the probability distribution of a probability model is as follows: To find the mean (AKA – expected value), multiply each possible value by its probability, then add all the products. Value of Xx1x1 x2x2 x3x3 xnxn Probabilityp1p1 p2p2 p3p3 pnpn

4. Calculator Shortcut EX: The distribution of the count of heads in 4 tosses was found to be: = 0(.0625) + 1(.25) + 2(.375) + 3(.25) + 4(.0625) = 2 Put X in L1 and P(X) in L2, in L3 (L1 x L2). You then can sum this total to get the mean or expected value. You can also run a 1VARS Stats on L1,L2 and it will produce the expected value.

5. Standard Deviation of a Discrete Probability Model If the probability distribution of a probability model is as follows: To find the standard deviation of the model: Value of Xx1x1 x2x2 x3x3 xnxn Probabilityp1p1 p2p2 p3p3 pnpn

6. Calculator: L1: Put the values of the random variable L2: Put the probabilities of each value At this point you can run a 1 VARS Stats : L1, L2 it will give you the expected value and the standard deviation.

7. Mean of a Continuous Probability Model What about continuous probability models? Think of the area under a density curve as being cut out of solid homogenous material. The mean μ is the point at which the shape would balance. This is what this idea looks like with a skewed model:

8. Mean of a Continuous Probability Model When the model is symmetric (normal, uniform, or other symmetric shape), the mean (and the median) lies at the center of the curve. Mean Median

9. The Law of Large Numbers

10. The law of large numbers tells us that in many trials the proportion of trials on which an outcome occurs will always approach its probability. The law of large numbers also explains why gambling can be a business. The winnings (or losses) of a gambler on a few plays are uncertain—that's why gambling is exciting. It is only in the long run that the mean outcome is predictable. The house plays many tens of thousands of times. So the house, unlike individual gamblers, can count on the long-run regularity described by the law of large numbers.

11. Homework Worksheet 8.5