This document discusses several continuous probability distributions including the exponential, gamma, uniform, and beta distributions. It provides the probability density functions and properties of each distribution. An example problem demonstrates calculating the probability that daily power consumption exceeds the plant's capacity using a gamma distribution.

1. Lecture 12: More Named
Continuous RV
2
1
Probability Theory and Applications
Fall 2008
October 9
σ 2
1
σ
All possible definitions of probability fall short of the actual practice.
William Feller

2. Outline
• Exponential Review
• Gamma
• Uniform
• Beta

3. PDF of named distributions
• Note you can use an applet to see what
happens when you change parameters of
the named distributions
http://www.causeweb.org/repository/statjava
/Distributions.html

4. Life Length Problem
Assume X = the life length in years of my
1998 Buick Park Avenue is an exponential
random variable with mean 10.
Given that the car more than14 years old,
what is the prob. that it will run more than
h years?
Under exponential assumption
P(X>14+h|X>14) =P(X>h)
The exponential is memoryless. Doesn’t seem like right distribution.

5. PDF under exponential Model
0 5 10 15 20 25 30 35 40
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
x
exponential with mean 10: exp(-x/10)/10

6. More Realistic PDF for car model
0 5 10 15 20 25 30 35 40
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
x
x8-1
exp(-x 8/10)/((8/10)8
)/γ(8)
0.25 1.25
8
1
0
( ) 1.25 7!
0 . .
x
x e x
f x
o w
−
⎧
>
⎪
= ⎨
⎪
⎩

7. Under new model
Assume X = the life length in years of my
1998 Buick Park Avenue has the pdf in the
previous slide (note mean is still 10)
Given that the car more than14 years old,
what is the prob. that it will run more than
h=2 years?
P(X>14+2|X>14) =0.4583
Which is must worse than
P(X>2)=0.9997
Clearly this is not a memory less distribution

8. Gamma Distribution
X has gamma distribution with parameters
k and θ if and only if X has pdf
1
1
0
1
e 0
( ) ( )
0 . .
( )
x
k
k
k t
x x
f x k
o w
with k t e dt
θ
θ
−
−
∞
− −
⎧
<
⎪
= Γ
⎨
⎪
⎩
Γ = ∫
Exponent is special case of Gamma with k=1

9. Gamma
Mean
in example mean=8*1.25=10
Variance
in example variance
=8*1.25*1.25=12.5
Note exponential is special case of gamma
with k=1
kθ
2
kθ

10. Meaning of parmaters
• Θ is the rate parameter
• k is the scale parameter
Has more effect on the variance/how
much distribution spreads out

11. Gamma generalizes factorial
Integrate by parts:
1 1 2
0
0
0
1 2
0
0
2
0
( ) ( 1)
( 1)
0 ( 1) ( 1) ( 1)
t
t t
t t
t
e t dt u t du t dt
uv vdu dv e v e
t e e t dt
e t dt
α α α
α α
α
α α
α
α α α
∞
− − − −
∞
∞ − −
∞
∞
− − − −
∞
− −
Γ = = = −
= − = = −
= − − − −
= + − − = − Γ −
∫
∫
∫
∫

12. Properties of Gamma
1. Generalizes factorial for α≥1
2.
3. If α=n is an integer >0
( ) ( 1) ( 1)
α α α
Γ = − Γ −
0
(1) 1
y
e dy
∞
−
Γ = =
∫
( ) ( 1) ( 1) ( 1)( 2) ( 2)
( 1)! (1)
( 1)!
n n n n n n
n
n
Γ = − Γ − = − − Γ −
= − Γ
= −

13. Problem
In a certain city, the daily consumption of
electric power in millions of kilowatt hours
can be treated as a random variable
having a gamma distribution with k=3 and
θ=2. If the power plant has a daily
capacity of 12 million kilowatt hours, what
is the probability that the power supply will
be inadequate on a given day?

14. Demand PDF
0 5 10 15 20 25 30
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
x
1/16 x2
exp(-x/2)
mean=2*3=6

15. Answer
The pdf is
Integrate to get probability
2 / 2
12
2 /2 / 2 / 2
12
6
1
( 12)
8*2
1
2 8 16
16
400 .062
t
t t t
P X t e dt
t e te e
e
∞
−
∞
− − −
−
≥ =
⎡ ⎤
= − − −
⎣ ⎦
= =
∫
2 2
3
1
e 0
( ) 2 (3)
0 . .
x
x x
f x
o w
−
⎧
= <
⎪
= Γ
⎨
⎪
⎩
Probability inadequate e.g.
Exceeds 12

16. Uniform Distribution
X ~ Uniform(a,b) a< b
Mean (a+b)/2 variance (b-a)2/12
What is cdf?
1
( )
0 . .
a x b
f x b a
o w
⎧
= < <
⎪
= −
⎨
⎪
⎩
a b

17. Examples of Uniform
• Alien abduction between mile marker 0
and 200. Uniform(0,200)
• X~Uniform(0,10)
Find
Rewrite as P(X2-7X+10 ≥0)
10
( 7)
P X
X
+ ≥

18. answer
X2-7X+10 has roots at 2 and 5
Want P(X≤2)+P(X≥5)=(2-0)/10+(10-5)/10
=7/10
2 5

19. Beta: Generalization of Uniform
Proportion of new restaurants failing in a
given city has the following pdf:
What is the probability at least 25% of the
restaurants will fail?
3
4(1 ) 0 1
( )
0 . .
x x
f x
o w
⎧ − = < <
= ⎨
⎩
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
2
2.5
3
3.5
4
x
4 (1-x)3
1
3
.25
( .25) 4(1 ) .3164
P X x dx
≥ = − =
∫

20. Beta: Generalization of Uniform
Beta with parameters α>0 β>0 lets us make
custom shaped distributions for RV
between 0 to 1
http://www.math.uah.edu/stat/special/Beta.x
html
1 1
1
1 1
0
(1 ) 0 1
( )
0 . .
( , ) (1 )
1
( , )
cx x x
f x
o w
Beta Integral x x
c
α β
α β
α β
α β
− −
− −
⎧ − = < <
= ⎨
⎩
Β = −
=
Β
∫

21. Facts about Beta
If α and β are integers,
Find c?
In general Mean = Variance =
( ) ( )
( , )
( )
α β
α β
α β
Γ Γ
Β =
Γ +
3 5
(1 ) 0 1
( )
0 . .
4 6
1 ( ) (10 1)!
504
( , ) ( ) ( ) (4 1)!(6 1)!
cx x x
f x
o w
Beta with
c
α β
α β
α β α β
⎧ − = < <
= ⎨
⎩
= =
Γ + −
= = = =
Β Γ Γ − −
α
α β
+ ( 1)( )
αβ
α β α β
+ + +
4
10
α
α β
=
+
4*10 2
(4 10 1)(1 10) 125
=
+ + +