1. Simulation Example: Generate a distribution for the random variate: What is the approximate probability that you will draw X ≤ 1.5?

2. Simulation

7. HHHHMMMM…

8. Simulation What if we crank up the sample sizes?

9. Simulation

12. Here is how we did all of that:

13. Simulation So what was the mean of that original Weibull distribution anyway? Looks like that’s where the mean on the histograms is going

14. Simulation How about the variance and sd? = 0.376  Weibull = 0.613  

15. So what did we observe?

16. Looks like the distribution of the sample means is a little skewed, but they’re approaching normal So what did we observe?

17. Generalizing to a IID sample, it can be shown that:

18. The Central Limit Theorem For a sum of IID random variables For an average of IID random variables

19. Example Random errors in the response of an instrument’s detector occur according to some unknown distribution, but have a mean = 2.45 units and variance = 0.63 units 2. Approximately, what is the probability that the cumulative error is greater that 123 units after 50 measurements? Plot the approximate probability density for the cumulative error after 73 measurements.

20. Simulation A a certain stock price moves with the approximate behavior from month to month: Pr(no change) = 0.5 Pr(up 5%) = 0.25 Pr(down 5%) = 0.25 Plot a realization of the stock’s movement assuming an initial stock price of $10. Approximately how much money to you expect to make/loose after one year of owning the stock? Three years? Ten years?