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

- Uses NORMINV function

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

Simulating Normal Random Variables

• Sample:

Number of columns

Number of rows

Type of distribution

Mean

Standard Deviation

(Leave blank)

Cell where data is placed

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

Simulating Normal Random Variables

• Soln:

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

Simulating Normal Random Variables

• Soln:

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)

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

Simulating Normal Random Variables

• Sample:

- Any number between

0 and 1 (Use RAND( )

for random values)

- Mean

- Standard Deviation

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

Simulating Normal Random Variables

• Soln:

Normal, CalculusNormal Distributions. Normal Distributions. CalculusCalculusNormal Distributions. Normal Distributions. CalculusCalculus

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, Calculus

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.

F X (x)

0.0

0.2

0.4

0.6

0.8

1.0

0 3 6 9 12 15

x

F X (x )DERIVATIVE OF F X (x)

0.00.10.20.30.40.50.6

0 3 6 9 12 15

x

F X ' (x )

Normal Distributions.Normal Distributions.Calculus: page 2Calculus: page 2

Normal Distributions.Normal Distributions.Calculus: page 2Calculus: page 2

f X (x)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 3 6 9 12 15

x

f X (x )

Normal, CalculusNormal Distributions. Normal Distributions. Calculus: page 3Calculus: page 3Normal Distributions. Normal Distributions. Calculus: page 3Calculus: 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.

.)()0(

0x

X duufxXP

duufxFx

XX )()(

0

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., 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. Normal Distributions. Calculus: page 4Calculus: page 4Normal Distributions. Normal Distributions. Calculus: page 4Calculus: page 4

,)(

0

duufx

X

duufx

X )(0

duufxFx

XX )()(

0

Normal, Calculus

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

Normal Distributions. Normal Distributions. Calculus: page 4Calculus: page 4Normal Distributions. Normal Distributions. Calculus: page 4Calculus: page 4

,)(

0

duufx

X

20/1)( xf X

Normal, Calculus

(material continues)

Normal Distributions. Normal Distributions. Calculus: page 6Calculus: page 6Normal Distributions. Normal Distributions. Calculus: page 6Calculus: 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.

,)( duufx

a

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.

,)()( duufzFz

ZZ

,)( duufz

Z

C IT

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.

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.

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.

Nash game