1. Simulating Normal Random Variables Simulation can provide a great deal of information about the behavior of a random variable

2. Simulating Normal Random Variables Two types of simulations (1) Generating fixed values - Uses Random Number Generation (2) Generating changeable values - Uses NORMINV function

3. Simulating Normal Random Variables Fixed Values Random Number Generation is found under Data/Data Analysis Values will never change Useful if you need to show how your specific results are tabulated

4. Simulating Normal Random Variables Sample: Number of columns Number of rows Type of distribution Mean Standard Deviation (Leave blank) Cell where data is placed

5. Simulating Normal Random Variables Ex. Generate a fixed sample of data containing 25 values that has a normal distribution with a mean of 13 and a standard deviation of 4.6

6. Simulating Normal Random Variables Soln:

7. Simulating Normal Random Variables Ex. Generate a fixed sample of data containing 30 values in cells A1:F5 that has a normal distribution with a mean of 81 and a standard deviation of 21

8. Simulating Normal Random Variables Soln:

9. Simulating Normal Random Variables(NORMINV-used in Project 2 Changeable Values NORMINV function Values will change by pressing F9 or by setting calculation in automatic mode Useful if you want several samples to average (eliminates a small number of poor values)

10. Simulating Normal Random Variables NORMINV will be used to run simulations for the project Sample data looks similar to Random Number Generation sample data Difference is that values can change by pressing F9

11. Simulating Normal Random Variables Sample: - Any number between 0 and 1 (Use RAND( ) for random values) - Mean - Standard Deviation

12. Simulating Normal Random Variables Ex. Generate a changeable sample of data containing 30 values in cells A1:F5 that has a normal distribution with a mean of 81 and a standard deviation of 21

13. Simulating Normal Random Variables Soln:

14. Normal, Calculus Normal Distributions. Calculus 4. Calculus* The Fundamental Theorem of Calculus, that gives a connection between the two main components of calculus, differentiation and integration, Let X be an exponential random variable with parameter  = 2. use Differentiating.xls to plot both F X (x) and its derivative for positive values of x. We also plot f X (x) for positive values of x.

15. Normal, Calculus It appears that, for positive values of x, the graphs of the p.d.f., f X, and the derivative, F X, of the c.d.f. are identical. Normal Distributions. Calculus: page 2 Normal Distributions. Calculus: page 2

16. Normal, Calculus Normal Distributions. Calculus: page 3 In summary, where the cumulative distribution function, F X, is differentiable, its derivative is the probability density function, f X. Hence, the c.d.f., F X, for the continuous exponential random variable, X, is the integral of the p.d.f., f X.

17. Normal, Calculus These relationships are not peculiar to exponential random variables. Let X be any continuous random variable.  The integral of the p.d.f., f X, is the c.d.f., F X.  Where F X is differentiable, its derivative is f X.  These can be combined to show that the derivative of with respect to x, is f X (x). Example:If X is a uniform random variable on the interval [0,20]. What is the derivative of Normal Distributions. Calculus: page 4

18. Normal, Calculus  These can be combined to show that the derivative of with respect to x, is f X (x). Example:If X is a uniform random variable on the interval [0,20]. What is the derivative of We know for uniform the p.d.f is a horizontal line between 0 and 20. here u=20, the Final Answer Normal Distributions. Calculus: page 4

19. Normal, Calculus (material continues)  Normal Distributions. Calculus: page 6 Fundamental Theorem of Calculus. For many of the functions, f, which occur in business applications, the derivative of with respect to x, is f(x). This holds for any number a and any x, such that the closed interval between a and x is in the domain of f. Combining this with our earlier information that we again see that the derivative of with respect to z, is f Z (z). This inverse relationship between integration and differentiation for probability functions is another instance of the Fundamental Theorem of Calculus, as stated previously in the section Calculus of Integration from Project 1. CIT

20. Nash equilibrium Definition What actions will be chosen by the players in a strategic game? We assume that each player chooses the action that is best for her, given her beliefs about the other players' actions. How do players form beliefs about each other? We consider here the case in which every player is experienced: she has played the game sufficiently many times that she knows the actions the other players will choose. Thus we assume that every player's belief about the other players' actions is correct.

21. Nash equilibrium A Nash equilibrium of a strategic game is an action profile (list of actions, one for each player) with the property that no player can increase her payoff by choosing a different action, given the other players' actions.

22. Nash equilibrium Note that nothing in the definition suggests that a strategic game necessarily has a Nash equilibrium, or that if it does, it has a single Nash equilibrium. A strategic game may have no Nash equilibrium, may have a single Nash equilibrium, or may have many Nash equilibria.

23. Nash game