Beta Distribution

Continuous Distributions Standard Normal Distribution PDF:

f  z   k exp   z 2 2  ,

k

1 2

1  2    exp   z 2   2  k  Mean = Mode: Variance:

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E Z   0 Var  Z   1

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Normal Distribution PDF:

 1  x   2  f  x |  ,    exp    ,  2        k

k

1 2

   1  x   2  dx   2     exp        2     k   

EX   

Mean = Mode:

Var  X    2

Variance

Uniform PDF:

1, 0    1 f     0, elsewhere

Mean:

E     1 2  0.5

Var     1 12  0.08333

Variance:

PDF

1

0 0

0.5

1

pi

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Beta Distribution   1 k  1 1    , 0    1  PDF: g    g  |  ,     , k  elsewhere  0,

1   ,  



    

      

1        1   1  1    ,        1    d  k      0

E  

Mean:

Var    

Variance:

  

, ,   0

 , ,   0         1 2

3.0

3.0

2.5

   1

2.0 PDF

PDF

2.5

    0.5

2.0 1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

X

0.6

0.7

0.8

0.9

1.0

X

3.0

4.0

2.5

3.5

  2

    12

3.0

PDF

2.0 PDF

0.5

1.5

2.5 2.0 1.5

1.0

1.0 0.5

0.5

0.0

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

X

0.5

0.6

0.7

0.8

0.9

1.0

0.6

0.7

0.8

0.9

1.0

X

12.0 10.0

PDF

PDF

3.0

  1 2,   2

8.0 6.0 4.0

  8,   3

2.0

1.0

2.0 0.0

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

X

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0.2

0.3

0.4

0.5 X

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Mode:

Mode    

 1 , ,   0   2

  x    x  1 !, x  1, 2,   x   ( x  1)  x  1 , x  .

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JMP Script for Beta Distribution

Figure 1. Beta (2, 2) PDF and CDF. Beta PDF and CDF https://courseware.vt.edu/users/holtzman/common/data/JMP_Scripts/BetaPDFandCDF.J SL

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Minitab Implementation for Bolstad Chapter 7, Example 12

Normal Approximation to the Beta PDF

Bolstad Chapter 7 Example 12 Beta(12, 25), Mean=0.3243, Var=0.005767, SD=0.07594 0.4 Variable Beta pdf Normal pdf

5

Y-Data

4 3 2 1 0 0.0

0.1

0.2

0.3 x

0.4

0.5

0.6

To approximate P(Y>0.4)

Normal Approximation to the Beta CDF

Bolstad Chapter 7 Example 12 Beta(12, 25), Mean=0.3243, Var=0.005767, SD=0.07594 0.4 1.0

Variable Beta(12,25) CDF Normal A pprox

Y-Data

0.8 0.6 0.4 0.2 0.0 0.1

0.2

0.3

0.4

0.5

0.6

x To approximate P(Y>0.4)

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The Minitab project that calculated and graphed this distribution is on line at http://courseware.vt.edu/users/holtzman/common/data/Examples/BetaNormalApproxBols tadCh7Example12.MPJ For Bolstad Chapter 7, Example 12, there is an error. The answer is not 0.3406. The correct answer using Bolstad’s style of Normal table that gives P(0 < Z < z) is P(Y > 0.4) = 0.5 – 0.3406 = 0.1594 Using Minitab, we find the normal approximation is P(Y > 0.4) = 0.15942 Using Minitab, we find the exact Beta-distribution value is P(Y > 0.4) = 0.16201 so the normal approximation was correct to two decimal places and two significant figures.

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Gamma Distribution g  x   g  x | r, v   kx r 1evx ,

PDF:

k  v r  r

1 r    r 1  vx    x e dx  r  v  k r E  X   , r, v  0 v

Mean:

Var  X  

Variance:

r v2

1.0

PDF

0.8

r=v=4

0.6 0.4 0.2 0.0 0

1

2

3

4

5

X

Gamma PDF and CDF JMP script If Y ~ Chi-Squared with d degrees of freedom, then Y ~ Gamma with r = d/2 and v = ½.

d 2 d 12 d 2 Var Y    2d 2 1 2  E Y  

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Exercises 1. Using Minitab, JMP, R, SAS, or whatever you like, for Y~Beta(1, 1) a. calculate and graph the PDF and CDF of Beta(1,1), b. calculate the mean, median, variance, standard deviation, and mode. 2. Repeat for Beta (0.5, 0.5). 3. Repeat for Beta (0.5, 2.0). 4. Repeat for Beta (2.0, 0.5). 5. Repeat for Beta (2, 2).

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