chapter 3 pp. 101-152
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Chapter 3 pp. 101-152. William J. Pervin The University of Texas at Dallas Richardson, Texas 75083. Chapter 3. Continuous Random Variables. Chapter 3. 3.1 Cumulative Distribution Function : The CDF F X of a RV X is F X (x) = P[X ≤ x] F X (-∞)=0; F X (+∞) = 1 - PowerPoint PPT PresentationTRANSCRIPT
The Erik Jonsson School of Engineering and Computer Science
Chapter 3pp. 101-152
William J. Pervin
The University of Texas at Dallas
Richardson, Texas 75083
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
Continuous Random Variables
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
3.1 Cumulative Distribution Function:
The CDF FX of a RV X is
FX(x) = P[X ≤ x]
FX(-∞)=0; FX (+∞) = 1
P[x1 < X ≤ x2] = FX(x2) – FX(x1)
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
Definition:
A RV X is continuous if its CDF FX is continuous
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
3.2 Probability Density Function:
The PDF fX of a continuous RV X is
fX(x) = dFX(x)/dx
Or, FX(x) = ∫-∞
x
fX(t)dt
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
For a continuous RV X with PDF fX(x):
(a) fX(x) ≥ 0 for all x
(b) FX(x) = ∫-∞x fX(u)du
(c) ∫-∞+∞ fX(x)dx = 1
Note: We do not require fX(x) ≤ 1
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
P[x1 < X ≤ x2] = ∫x1
x2 fX(x)dx
Note that endpoints don’t matter!
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
3.3 Expected Values:
E[X] = ∫ xfX(x)dx = μX
E[g(X)] = ∫ g(x)fX(x)dx
Var[X] = ∫ (x - μX)2 fX(x) dx
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
E[X – μX] = 0; that is, μX = E[X]
E[aX + b] = aE[X] + b
Var[X] = E[X2] – μX2
Var[aX + b] = a2Var[X]
Thus, σaX+b = aσX
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
3.4 Families of Continuous RVs:
Uniform (a,b):
fX(x) = 1/(b-a) if a≤x<b, 0 otherwise
FX(x) = 0 , x ≤ a
= (x-a)/(b-a) , a < x ≤ b
= 1 , x > b
E[X] = (b+a)/2; Var[X] = (b-a)2/12
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
Exponential (λ):
fX(x) = λe-λx, x ≥ 0, 0 otherwise PDF
FX(x) = 1 – e-λx, x ≥ 0, 0 otherwise CDF
E[X] = 1/λ Var[X] = 1/λ2 σX = 1/λ
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
If K = ┌X┐ then:
If X is uniform (a,b) with a,b integers, then K is discrete uniform (a+1, b).
If X is exponential (λ) then K is geometric (p = 1 - e-λ).
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
3.5 Gaussian RVs:
Gaussian (μ,σ):
fX(x) = (2πσ2)-1/2 exp{-(x-μ)2/2σ2}
E[X] = μ; Var[X] = σ2 ; [S.D. = σ]
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
Theorem: If X is Gaussian (μ,σ) then
Y = aX + b is Gaussian (aμ + b, aσ).
Standard Normal RV Z is Gaussian (0,1)
Standard Normal CDF
ΦZ(z) = (2π)-1/2 Int{e-t2/2dt,-∞,z}
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
If X is Gaussian (μ,σ) RV, the CDF of X is
FX(x) = Φ((x-μ)/σ)
P[a < X ≤ b] = Φ((b-μ)/σ) – Φ(a-μ)/σ)
Tables use z = (x-μ)/σ standard deviations from the mean
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
For negative values in the tables use
Φ(-z) = 1 – Φ(z)
Standard Normal Complementary CDF
Q(z) = P[Z > z] = 1 – Φ(z)
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
3.6 Delta Functions; Mixed RVs:
Unit impulse (Delta) “function” δ has the property that, for any continuous g(x):
Int{g(x)δ(x-x0)dx,-∞,+ ∞} = g(x0)
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
Unit step function u:u(x) = 0, x < 0 = 1, x ≥ 0
u(x) = Int{δ(t)dt,-∞,x}δ(x) = du(x)/dx
Mixed RVs contain impulses and values
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
3.7 Probability Models for Derived RVs:
If Y = g(X), how to determine fY(y) from g(X) and fX(x):
1. Find CDF FY(y) = P[Y≤y]
2. Take derivative fY(y) = dFY(y)dy
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
Let U be uniform (0,1) RV and let F be a CDF with inverse F-1 defined on (0,1). The RV X = F-1(U) has CDF FX(x)=F(x).
Note: Most random number generators yield the uniform (0,1) distribution.
This method is very important for simulation work with other distributions!
The Erik Jonsson School of Engineering and Computer Science
Chapter 3
3.8 Conditioning a Continuous RV
3.9 MATLAB