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Simulating Normal Random Variables - PowerPoint PPT Presentation

Simulating Normal Random Variables. Simulation can provide a great deal of information about the behavior of a random variable. Simulating Normal Random Variables. Two types of simulations (1) Generating fixed values - Uses Random Number Generation (2) Generating changeable values

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Simulating Normal Random Variables

Simulation can provide a great deal of information about the behavior of a random variable

• Two types of simulations

(1) Generating fixed values

- Uses Random Number Generation

(2) Generating changeable values

- Uses NORMINV function

• 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

• Sample:

Number of columns

Number of rows

Type of distribution

Mean

Standard Deviation

(Leave blank)

Cell where data is placed

• 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

• 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

• 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)

• 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

• Sample:

- Any number between

0 and 1 (Use RAND( )

for random values)

- Mean

- Standard Deviation

• 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

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 FX(x) and its derivative for positive values of x. We also plot fX(x) for positive values of x.

Normal Distributions.

Calculus: page 2

It appears that, for positive values of x, the graphs of the p.d.f., fX, and the derivative, FX, of the c.d.f. are identical.

Normal Distributions. Calculus: page 3

In summary, where the cumulative distribution function, FX, is differentiable, its derivative is the probability density function, fX.

Hence, the c.d.f., FX, for the continuous exponential random variable, X, is the integral of the p.d.f., fX.

Normal Distributions. Calculus: page 4

These relationships are not peculiar to exponential random variables. Let X be any continuous random variable.

The integral of the p.d.f., fX, is the c.d.f., FX.

Where FX is differentiable, its derivative is fX.

These can be combined to show that the derivative of

with respect to x, is fX(x).

Example:If X is a uniform random variable on the interval [0,20]. What is the derivative of

Normal Distributions. Calculus: page 4

• These can be combined to show that the derivative of

• with respect to x, is fX(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

Combining this with our earlier information that

we again see that the derivative of with respect to z, is fZ(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.

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.

Normal Distributions. Calculus: page 6

T

C

I

(material continues)

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.

• ANash equilibriumof 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,giventhe other players' actions.

• 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.