the erik jonsson school of engineering and computer science chapter 3 pp. 101-152 william j. pervin...

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


Chapter 3

Continuous Random Variables


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)


Chapter 3

Definition:

A RV X is continuous if its CDF FX is continuous


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


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


Chapter 3

P[x1 < X ≤ x2] = ∫x1

x2 fX(x)dx

Note that endpoints don’t matter!


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


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


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


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


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).


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]


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}


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


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)


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)


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


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


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!


Chapter 3

3.8 Conditioning a Continuous RV

3.9 MATLAB

the erik jonsson school of engineering and computer science chapter 3 pp. 101-152 william j. pervin...

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