Download Simulating Normal Random Variables

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 Distributions. Calculus
Normal, 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
Normal
Calculus: page 2
F X (x)
1.0
0.8
F X (x )
0.6
DERIVATIVE OF F X (x)
0.4
0.2
0.0
0
3
6
9
12
15
x
0.6
0.5
0.4
F X ' (x ) 0.3
0.2
0.1
0.0
0
f X (x)
3
6
9
12
x
0.6
0.5
0.4
f X (x ) 0.3
0.2
0.1
0.0
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.
0
3
6
9
x
12
15
15
Calculus page 3
Normal Distributions.Normal,
Calculus:
In summary, where the cumulative distribution function, FX, is
differentiable, its derivative is the probability density function, fX.
x
P(0  X  x ) 

f X (u ) du.
0
Hence, the c.d.f., FX, for the continuous exponential random variable, X, is the
integral of the p.d.f., fX.
Calculus page 4
Normal Distributions.Normal,
Calculus:
x
FX ( x ) 
 f X ( u) du
0
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.
x
 These can be combined to show that the derivative of
 f X (u) du,
0
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
x
f
0
X
(u ) du
Calculus page 4
Normal Distributions.Normal,
Calculus:
x
FX ( x ) 
 f X ( u) du
0
 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
x
 f X (u) du,
0
•We know for uniform the p.d.f is a horizontal line between 0 and 20. here
u=20, the Final Answer
f X ( x)  1/ 20
Calculus page 6
Normal Distributions.Normal,
Calculus:
z
Combining this with our earlier information that FZ ( z ) 
z
we again see that the derivative of
 f Z (u) du,

 f Z (u) du, 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.
Fundamental Theorem of Calculus. For many of the functions, f,
x
which occur in business applications, the derivative of
 f (u) du, with
a
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.
(material continues)

T
C
I

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