1. Lecture (10) Mathematical Expectation

2. The expected value of a variable is the value of a descriptor when averaged over a large number theoretically infinite.

3. Mathematical Expectation (cont.) Another way to compute the variance

4. Example 1 TTTHHTHH 012 Sample SpaceNumber of Heads

5. Example 1 (cont.)

6. 3210 1 0.5.25 NUMBER OF HEADS P R O B A B I L I T Y Experiment: Toss Two Coins

7. Example 1 (cont) E.G. Toss 2 coins, count heads, compute expected value:  = 0 .25 + 1 .50 + 2 .25 = 1 E.G. Toss 2 coins, count heads, compute variance: variance = (0 - 1) 2 (.25) + (1 - 1) 2 (.50) + (2 - 1) 2 (.25) =.50

8. Example 2

9. Find the mean of the number of spots that appear when a die is tossed. The probability distribution is given below. Discrete Uniform Distribution Example

10. That is, when a die is tossed many times, the theoretical mean will be 3.5. That is, when a die is tossed many times, the theoretical mean will be 3.5. Discrete Uniform Distribution Example (cont.)

11. Binomial Distribution - Binomial Distribution - Example A coin is tossed four times. Find the mean, variance and standard deviation of the number of heads that will be obtained. Solution:Solution: n = 4, p = 1/2 and q = 1/2.  = n  p = (4)(1/2) = 2.  2 = n  p  q = (4)(1/2)(1/2) = 1.  = = 1.

12. Poisson Distribution

13. Uniform Distribution Example

14. If the probability density function has the form f(x) = ax for a random variable X between 0 and 2. (a)Find the value of a. (b) Find the median of X (c)Find P(1.0 < X < 2.0) Solution: (a) From the area under the whole density curve is 1, then we have Example

15. Quiz

16. Exponential Distribution

17. DistributionNormal x LogN Y =logx Gamma x Exp t Mean xx yy nk1/k Variance xx yy nk  1/k 2 Skewnesszero 2/n 0.5 2 Comparison of Parameters of Dist’n