1. 4.4 Mean and Variance

2. Mean How do we compute the mean of a probability distribution? Actually, what does that even mean? Let’s look at an example on the board. This leads us to define: Given a discrete random variable x with probability distribution P(x), the mean of the random variable x (or P(x)) is defined as

3. Expected Value Another name for the mean in this case is the expected value. For a random variable x, we define the expected value of the random variable, denoted E(x), to be E(x)=μ. Find the expected value of 0123.3.4.2.1

4. Practice Problem Find the expected value of the loaded die from section 4.2. 123456 2/151/6 1/5

5. Law of Large Numbers We have hinted at this before, but now we “officially” state the law of large numbers. It states that as the number of observations increases, the mean of the observed value eventually approaches the mean of the population as close as you would like. Thus, in the long run, casinos that have the odds in their favor will make money, for example.

6. A card is to be selected from an ordinary deck of 52 cards. Suppose the casino will pay $10 if you select an ace. If you fail, you pay the casino a dollar. How much does the casino expect to win per game on average if you play this game many times?

7. Suppose we are studying days of scheduled classes missed in Colleges and Universities in Minnesota. We find that on average there are 1.7 days missed due to the teacher cancelling class and 4.6 days missed due to the school cancelling because of weather. How many total days are missed on average in Minnesota? 1.7+4.6= 6.3

8. Rules for Mean We have illustrated the principle that if X is random variable and Y is another random variable, then X+Y is also a random variable. We also have the formula μ X+Y = μ X +μ Y. Let a, b be fixed numbers. Another formula is μ a+bX = a + bμ X. This formula says that if I multiply or add a constant to the random variable, then the mean changes accordingly.

9. Practice Problems The random variable X has mean μ X =5. If Y=3-2X, what is μ X ? The expected payoff for a game is $4.00. If the random variable X is in terms of dollars, convert it to cents and find the expected payoff for that same game in cents, justifying your answer.

10. Variance There is a similar formula for variance. We have: Given a discrete random variable x with probability distribution P(x), the variance of the random variable x (or P(x)) is defined as We then define the standard deviation as the square root of the variance. Notice that there is no concept of population vs. sample.

11. Example Consider the data set Compute the mean, variance, and standard deviation. xP(x) 12/15 210/15 32/15 41/15

12. Note: An easier formula for variance is given by Now compute the variance for the above data using this formula. Wow, much easier!

13. Independence and Correlation Two random variables X and Y are independent if knowing that any event involving X alone did or did not occur tells us nothing about the occurrence of any event involving Y alone. An example is the rolling of two dice. Dealings in a game of blackjack is an example of dependence of events. We may associate a correlation ρ (rho) between two random variables. This is similar to the correlation r we studied in chapter 2. The correlation between two independent random variables is 0.

14. Example Scores on the Math part of the SAT have a mean of 519 and standard deviation of 115. Scores on the verbal part had mean 507 and standard deviation 111. 1. What is the mean of total SAT scores? 2.What is the standard deviation of total SAT scores? 3.Need correlation ρ =.71

15. Practice Problem The amount of profit for a major investment is uncertain, but a probabilistic estimate gives the following distribution (in millions of dollars). Find the mean profit and the standard deviation of the profit. The investment firm owes its source of capital a fee of $200,000 plus 10% of the profits. The firm thus retains Y=.9X-.2 from the investment. Find the mean and standard deviation of Y. Profit11.52410 Probability.4.2..2.1