Download 3. Analytical Models of Random Phenomena

3. Analytical Models of Random Phenomena
3.1 Random Variables and Probability Distribution
3.2 Useful Probability Distributions
3.3 Multiple Random Variables
3.4 Concluding Summary
3.1 Random Variables and Probability Distribution
3.1.1 Random Events and Random Variables
Random Variable:
An event may be identified through the values of a function
Sample Space S
E1
E1 = ( a < X ≦ b)
E2 = ( c < X ≦ d)
E2
Random Variable X
E1 U E 2
a
c
b
d
E1E2
A rule that maps events in a sample space into the real line
3.1.2. Probability Distribution of a Random Variable
Probability Distribution:
The rule for describing the probability measures associated
with all the values of a random variable
Discrete Random Variable:
Probability is defined for certain discrete values of x
Probability Mass Function (PMF)
Cumulative Distribution Function (CDF)
Continuous Random Variable:
Probability is defined for any value of x
Probability Density Function (PDF)
Cumulative Distribution Function (CDF)
X: Discrete
Probability Mass Function (PMF)
pX(x)
pX (xi) = P(X= xi)
1/6
0 ≤ pX ( xi ) ≤ 1
∑ p( x ) = 1
i
1
2
3
4
5
6
xi
Cumulative Distribution Function (CDF)
FX(x)
Fx(x) = P(X ≤ x)
1
=
∑
all xi ≤ x
1
2
3
4
5
6
(3.1)
pX (xi ) (3.2)
X: Continuous
Probability Density Function (PDF)
fX(x)
dFX ( x) (3.5)
f X ( x) =
dx
a
0
b
x
f ( x) ≥ 0
X
∫
∞
−∞
f X ( x)dx = 1
Cumulative Distribution Function (CDF)
FX(x)
1
x
Fx(x) = P( X ≤ x) = ∫ f x ( x)dx
−∞
(3.4)
b
Fx(b)
Fx(a)
0
P(a < X ≤ b) = ∫ f X (x)dx
a
x
(3.3)
= FX (b) − FX (a) (3.6)
FX (−∞) = 0
FX (∞) = 1
CDF
PMF PX(x)
FX(x)
PDF fX(x)
FX(x)
dFX ( x )
f X ( x) =
dx
Continuous Random Variable
Probability Density Function (PDF)
Cumulative Distribution Function (CDF)
Probabilty
Probabilty
1
dx
Discrete Random Variable
Probability Mass Function (PMF)
Cumulative Distribution Function (CDF)
Probabilty
1
Probabilty
1/6
1 2
3
4
5
6
1 2 3 4 5 6
PDF & CDF of Normal Distribution
PDF
CDF
3.1.3 Main Descriptors of a Random Variable
1. Central Value
Mean (Expected Value), Mode, Median
2. Measure of Dispersion
Variance, Standard Deviation,
Coefficient of Variance
3. Skewness Measure
Skewness Coefficient
1. Central Value
Mean or Expected Value
µx = E(X ) =
µx = E(X ) =
∑
all xi
∞
∫
−∞
xi p X ( xi )
(3.7a)
xf X ( x ) d x
(3.7b)
Mathematical Expectation:
Mode
µ x = E[ g ( X )] = ∑ g ( X i ) p (X i )
X
(3.9a)
µ x = E[ g ( X )] =
g ( X ) f X ( X ) dX
(3.9b)
all X i
∞
∫− ∞
~
x : the most probable value of a random variable
Median xm: the value of a random variable at which values
above and below it are equally probable
Fx (xm ) = 0.5
(3.8)
Example: Household Income Distribution
2008 in Japan
Mode
Mean
Median
Log-Normal Distribution
http://www.mhlw.go.jp/toukei/saikin/hw/k-tyosa/k-tyosa08/2-2.html
[million yen/household/year]
2. Measure of Dispersion
2
(
x
−
µ
)
∑ i x px (xi )
Variance: Var ( X ) = E[(
X − µx )2 ] =
(3.10a)
all xi
∞
= ∫ ( x − µx )2 f ( x)dx
−∞
Var ( X ) = E ( X 2 ) − µ x 2
(3.10b)
(3.11)
Standard Deviation:
σ x = Var ( x ) = E( X 2 ) − µ x 2
Coefficient of Variation (COV):
σx
COV =
µx
(3.12)
(3.13)
Coefficient of Variation
Month
January
February
March
April
May
June
July
August
September
October
November
December
Wine
8.9
9.1
10.6
9.2
9.0
7.6
8.8
7.5
6.9
9.8
10.0
25.2
Beer
13.8
16.2
20.3
24.3
28.3
28.4
36.8
33.6
22.8
20.9
19.8
32.2
[liter/household]
Mean
Wine 10.22
Beer 24.78
SD
Wine 4.63
Beer 6.84
COV
Wine 0.453
Beer 0.276
Effect of varying parameters
(µ & σ)
µ ↑ for C
fX(x)
B
σ ↓ for B
C
A
x
Characteristics of Mean and Variance
E(aX+b) = aE(X) + b
E(X+Y) = E(X) + E(Y)
E(XY) = E(X) E(Y) X, Y: Independent
Var(X)
= E(X2) – {E(X)} 2
Var(aX+b) = a2 Var(X)
f(X)
b
f(X+b)
Var(X+b) = Var(X)
X0
X0+b
f(aX)
f(X)
X0 µ+σ
aX0
aµ+aσ
µ x = E [ g ( X )] = ∫−∞∞ g ( X ) f X ( X ) dX
E(aX+b)
= ∫ (aX+b) f(X) dX
= ∫ aX f(X) dX + ∫b f(X) dX
= a ∫X f(X) dX + b∫ f(X) dX
= a E(X) +b
E(X+Y)
=∬(X+Y) f (X,Y) dYdX
=∬X
=
X f (X,Y) dYdX+
dYdX+∬Y
Y f (X,Y) dYdX
=∫X f X(X) dX+∫ Y fY (Y) dY
= E(X) +E(Y)
E(XY)
= ∬XY f (X,Y) dYdX
= ∬XY f X(X) fY (Y) dYdX
= ∫X f X(X) dX ･∫Y fY (Y) dY
= E(X) E(Y)
X, Y: Independent
Var(X)
= E[{X-E(X)}2]
= E [X 2 -2XE(X)+ { E(X) }2]
= E(X 2) –2E(XE(X))+E({ E(X) }2)
= E(X 2) –2 E(X) E(X)+ { E(X) }2
= E(X 2) –{ E(X) }2
Var(aX+b)
= E [{ aX+b - E(aX+b) }2]
= E [{ aX+b - aE(X) -b) }2]
= E [a2{ X- E(X) }2]
= a2 E [{ X- E(X) }2]
= a2 Var(X)
Ex. 3.5
fT(t): PDF of tolerance period of welding machine
f T ( t ) = λ e − λt
FT ( t ) =
∫
t
0
λe
(t ≥ 0)
− λτ
d τ = [− e ]
− λx t
0
∞
−λt
= − e − λ t − ( − 1) = 1 −e − λ t
∞
−λt ∞
0
1
−λt
−λt ∞
0
1
µT = E(T) = ∫ t ⋅ λe dt = −[te ] + ∫ e dt = − [e ] =
0
0
λ
λ
Var ( T ) =
σ
2
∫
∞
0
=
λ∫
=
1
∞
0
λ
t ⋅ λe
2
t ⋅ λe
− λt
− λt
dt − µ = −[ t e
2
T
2
∞
] + ∫ 2 t ⋅ e − λ t dt − µ T2
− λt ∞
0
0
2 1
1 2
1
dt − µ = ( ) − ( ) = 2
Median?
2
T
λ λ
λ
λ
Ex. 3.6
P: Probability of projects being completed as schedule
X: Number of jobs completed among 6 future jobs
P(X=x)=px(xi)=6Cxpx(1-p)6-x
6
6
x =0
x =0
x=0, 1, 2,…., 6
µ x = E(X) = ∑ xp x ( x ) = ∑ x⋅6 C x 0.6 x 0.46− x
=16 C1 0.610.45 + 26 C 2 0.620.4 4 +36 C3 0.630.43
+ 46 C 4 0.6 40.42 +56 C5 0.650.41 +66 C6 0.66 = 3.6
6
6
Var ( X ) = ∑ x p x ( x) − µ =∑ x 6 C x 0.6 0.4
2
x =0
2
x
2
x
6− x
−µ
2
x
0
1
2
x =0
= 1.44
Var(X)
1.44
COV =
=
= 0.3333
µx
3.6
PMF
Mode = 4
0.3500
0.3000
0.2500
0.2000
0.1500
0.1000
0.0500
0.0000
3
4
5
6
3. Measure of Skewness
Asymmetry =
Third Central Moment
E ( X − µ X )3 = ∑ ( X i − µ X )3 p X (X i )
all X i
E ( X − µ X ) 3 = ∫−∞∞ ( x − µ X ) 3 f X ( X ) dX
Skewness Coefficient:
E[( X − µ x ) 3 ]
θ=
σ 3x
(3.14)