Plots and Random #s
EXCEL Functions
Obtaining a Density Function
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Create a column with a range of values of
x containing a large portion of the density
Create a column that evaluates f(x) at
each value of x (Specific densities below)
Repeat previous step for multiple pdf’s
Highlight columns with x (left-hand
column) and f(x) (right hand column(s))
Chart Wizard  XY (Scatter)  Picture
with Smooth curves/No points
Families of Densities
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Gamma(a,b): f(x)=GAMMA.DIST(x,a,b,0)
• Exponential (a=1,q=1/b) special case of Gamma
• Chi-Square (a=n/2, b=2) special case of Gamma
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Beta(a,b): f(x)=BETA.DIST(x,a,b,0)
Normal(m,s2): f(x)=NORM.DIST(x,m,s,0)
t(n): f(x) = T.DIST(x,n,0)
F(n1,n2): f(x) = F.DIST(x,n1,n2,0)
Other densities can be obtained by
directly typing in f(x)
Example: Gamma(a=2,b=2)
Cell A1
Cell A1500
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.002488
0.00495
0.007388
0.009802
0.012191
0.014557
0.016898
0.019216
0.02151
…
…
14.95
14.96
14.97
14.98
14.99
15.00
0.002119
0.00211
0.002101
0.002092
0.002083
0.002074
=GAMMADIST(a1,2,2,0)
=GAMMADIST(a1500,2,2,0)
Example: Gamma(a=2,b=2)
Gamma(2,2) pdf
0.2
0.18
0.16
0.14
f(x)
0.12
0.1
0.08
0.06
0.04
0.02
0
0
2
4
6
8
x
10
12
14
16
Cumulative Distribution Function
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Create a column with a range of values of
x containing a large portion of the density
Create a column that evaluates F(x) at
each value of x (Specific denities below)
Repeat previous step for multiple pdf’s
Highlight columns with x (left-hand
column) and F(x) (right hand column(s))
Chart Wizard  XY (Scatter)  Picture
with Smooth curves/No points
Families of CDF’s
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Gamma(a,b): F(x)=GAMMADIST(x,a,b,1)
• Exponential (a=1) special case of Gamma
• Chi-Square (a=n/2, b=2) special case of Gamma
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Normal(m,s2): F(x)=NORMDIST(x,m,s,1)
Beta(a,b): F(x)=BETA.DIST(x,a,b,1)
t(n): F(x) = T.DIST(x,n,1)
F(n1,n2): F(x) = F.DIST(x,n1,n2,1)
Others obtained by directly entering cdf
Example - Gamma(a=2,b=2)
Cell A1
Cell A1500
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
1.25E-05
4.97E-05
0.000111
0.000197
0.000307
0.000441
0.000598
0.000779
0.000983
…
…
14.95
14.96
14.97
14.98
14.99
15.00
0.995194
0.995215
0.995236
0.995257
0.995278
0.995299
=Gammadist(a1,2,2,1)
=Gammadist(a1500,2,2,1)
Example - Gamma(a=2,b=2)
Gamma(2,2) CDF
1.2
1
F(x)
0.8
0.6
0.4
0.2
0
0
2
4
6
8
x
10
12
14
16
Selecting Pseudo-Random
Variables
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Select a sample size and density function
In EXCEL, type: =RAND()
Copy and Paste that cell to n-1 below it
Highlight all n cells:
• COPY  PASTE SPECIAL  VALUES
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This is a pseudo-random sample from a
uniform(0,1) distribution
Obtain random sample from inverting cdf
Families of Distributions
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Labelling your U(0,1) value as p
Gamma(a,b) X=GAMMA.INV(p,a,b)
Normal(m,s2) X=NORM.INV(p,m,s)
Beta(a,b) X=BETA.INV(p,a,b)
t(n)
X = T.INV(p)
F(n1,n2)
X = F.INV(x,n1,n2)
Copy and Paste this function for all n p’s
Simulating 10000 Gamma(2,2)
RVs
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In cell A1 type: =RAND()
Copy/Paste to cells A2:A10000
Copy/Paste Special/Values cells
A1:A10000
In cell B1 type: =GAMMAINV(A1,2,2)
Copy/Paste to cells B2:B10000 (May
take a few seconds)
Example 10000 Gamma(2,2)
1st 10 p
0.281723
0.681206
0.334421
0.555336
0.161319
0.521112
0.052864
0.39327
0.519182
0.10049
2.095076
4.706185
2.383676
3.723865
1.431617
3.493365
0.733489
2.714214
3.480709
1.066755
Empirical Results
Mean
Variance
n
p
0.5023
0.0839
10000
X
4.0311
8.2851
10000
1st 10 X
Theoretical Values
Mean
Variance
p
0.5000
0.0833
X
4.0000
8.0000
x
15
.5
13
12
.5
10
9
7.
5
6
4.
5
3
1.
5
1000
900
800
700
600
500
400
300
200
100
0
0
Frequency
10000 Gamma(2,2) RVs