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MISC.
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:
Integration Applications-oct1st
•
FundamentalTheorem
Theorem ofof
Calculus.
For many of the functions, f,
Fundamental
Calculus
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.
Example : applies to p.d.f.’s and c.d.f.’s
Recall from Math 115a

b
a
f X x  dx  FX b   FX a 
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Integration, Calculus
the inverse connection between integration and differentiation is
called the Fundamental Theorem of Calculus.
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.
Example 7. Let f(u) = 2 for all values of u. If x  1, then
integral of f from 1 to x is the area of the region over the interval [1, x],
between the u-axis and the graph of f.
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Integration, Calculus
3
(1, 2)
2
(x, 2)
The region whose area
is represented by the integral is
rectangular, with height 2 and
width x  1. Hence, its area is
2(x  1) = 2x  2, and
x
f (u )
1
 f (u) du
2
1
x1
0
x
0
1
2
3
u
x 4
5
 f (u) du  2  x  2.
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In the section Properties and Applications of Differentiation, we saw
that the derivative of f(x) = mx + b is equal to m, for all values of x. Thus, the
x
derivative of
 f (u) du, with respect to x, is equal to 2. As predicted by the
1
Fundamental Theorem of Calculus, this is also the value of f(x).
The next example uses the definition of a derivative as the limit of
difference quotients.
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