lecture 12: more named continuous rv - rensselaer...
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Lecture 12: More Named Continuous RV 2
1
Probability Theory and ApplicationsFall 2008October 9
σ 21σ
All possible definitions of probability fall short of the actual practice. William Feller
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
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.
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
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 1.258
1 0( ) 1.25 7!0 . .
x
x e xf xo w
−⎧>⎪= ⎨
⎪⎩
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) =0.4583Which is must worse than P(X>2)=0.9997
Clearly this is not a memory less distribution
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 . .
( )
xk
k
k t
x xf x k
o w
with k t e dt
θ
θ−−
∞− −
⎧<⎪= Γ⎨
⎪⎩
Γ = ∫Exponent is special case of Gamma with k=1
Gamma
Meanin 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
2kθ
Meaning of parmaters
• Θ is the rate parameter • k is the scale parameter
Has more effect on the variance/how much distribution spreads out
Gamma generalizes factorial
Integrate by parts:
1 1 2
0
00
1 2
00
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 −
∫
∫
∫
∫
Properties of Gamma
1. Generalizes factorial for α≥1
2.
3. If α=n is an integer >0
( ) ( 1) ( 1)α α αΓ = − Γ −
0
(1) 1ye dy∞
−Γ = =∫
( ) ( 1) ( 1) ( 1)( 2) ( 2)( 1)! (1)( 1)!
n n n n n nnn
Γ = − Γ − = − − Γ −= − Γ= −
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?
Answer
The pdf is
Integrate to get probability2 / 2
12
2 / 2 / 2 / 2
12
6
1( 12)8*2
1 2 8 1616400 .062
t
t t t
P X t e dt
t e te e
e
∞−
∞− − −
−
≥ =
⎡ ⎤= − − −⎣ ⎦
= =
∫
2 23
1 e 0( ) 2 (3)
0 . .
x
x xf x
o w
−⎧= <⎪= Γ⎨
⎪⎩
Probability inadequate e.g.Exceeds 12
Uniform Distribution
X ~ Uniform(a,b) a< b
Mean (a+b)/2 variance (b-a)2/12What is cdf?
1( )
0 . .
a x bf x b a
o w
⎧ = < <⎪= −⎨⎪⎩
a b
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 XX
+ ≥
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?
34(1 ) 0 1( )
0 . .x x
f xo 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
13
.25
( .25) 4(1 ) .3164P X x dx≥ = − =∫
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.xhtml
1 1
11 1
0
(1 ) 0 1( )
0 . .
( , ) (1 )
1( , )
cx x xf x
o w
Beta Integral x x
c
α β
α βα β
α β
− −
− −
⎧ − = < <= ⎨⎩
Β = −
=Β
∫
Facts about Beta
If α and β are integers,
Find c?
In general Mean = Variance =
( ) ( )( , )( )α βα βα β
Γ ΓΒ =
Γ +
3 5(1 ) 0 1( )
0 . .4 6
1 ( ) (10 1)! 504( , ) ( ) ( ) (4 1)!(6 1)!
cx x xf x
o wBeta with
c
α βα β
α β α β
⎧ − = < <= ⎨⎩
= =Γ + −
= = = =Β Γ Γ − −
αα β+ ( 1)( )
αβα β α β+ + +
410
αα β
=+
4*10 2(4 10 1)(1 10) 125
=+ + +
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