The Erik Jonsson School of Engineering and Computer Science Chapter 3 pp 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 Cumulative Distribution Function: The CDF F X of a RV X is F X (x) = P[X ≤ x] F X (-∞)=0; F X (+∞) = 1 P[x 1 < X ≤ x 2 ] = F X (x 2 ) – F X (x 1 )

The Erik Jonsson School of Engineering and Computer Science Chapter 3 Definition: A RV X is continuous if its CDF F X is continuous

The Erik Jonsson School of Engineering and Computer Science Chapter Probability Density Function: The PDF f X of a continuous RV X is f X (x) = dF X (x)/dx Or, F X (x) = ∫ -∞ x f X (t)dt

The Erik Jonsson School of Engineering and Computer Science Chapter 3 For a continuous RV X with PDF f X (x): (a) f X (x) ≥ 0 for all x (b) F X (x) = ∫ -∞ x f X (u)du (c) ∫ -∞ +∞ f X (x)dx = 1 Note: We do not require f X (x) ≤ 1

The Erik Jonsson School of Engineering and Computer Science Chapter 3 P[x 1 < X ≤ x 2 ] = ∫ x 1 x 2 f X (x)dx Note that endpoints don’t matter!

The Erik Jonsson School of Engineering and Computer Science Chapter Expected Values: E[X] = ∫ xf X (x)dx = μ X E[g(X)] = ∫ g(x)f X (x)dx Var[X] = ∫ (x - μ X ) 2 f X (x) dx

The Erik Jonsson School of Engineering and Computer Science Chapter 3 E[X – μ X ] = 0; that is, μ X = E[X] E[aX + b] = aE[X] + b Var[X] = E[X 2 ] – μ X 2 Var[aX + b] = a 2 Var[X] Thus, σ aX+b = aσ X

The Erik Jonsson School of Engineering and Computer Science Chapter Families of Continuous RVs: Uniform (a,b): f X (x) = 1/(b-a) if a≤x<b, 0 otherwise F X (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): f X (x) = λe -λx, x ≥ 0, 0 otherwise PDF F X (x) = 1 – e -λx, x ≥ 0, 0 otherwise CDF E[X] = 1/λ Var[X] = 1/λ 2 σ X = 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 Gaussian RVs: Gaussian (μ,σ): f X (x) = (2πσ 2 ) -1/2 exp{-(x-μ) 2 /2σ 2 } E[X] = μ; Var[X] = σ 2 ; [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 Φ Z (z) = (2π) -1/2 Int{e -t 2 /2 dt,-∞,z}

The Erik Jonsson School of Engineering and Computer Science Chapter 3 If X is Gaussian (μ,σ) RV, the CDF of X is F X (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 Delta Functions; Mixed RVs: Unit impulse (Delta) “function” δ has the property that, for any continuous g(x): Int{g(x)δ(x-x 0 )dx,-∞,+ ∞} = g(x 0 )

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 Probability Models for Derived RVs: If Y = g(X), how to determine f Y (y) from g(X) and f X (x): 1. Find CDF F Y (y) = P[Y≤y] 2. Take derivative f Y (y) = dF Y (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 F X (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 Conditioning a Continuous RV 3.9 MATLAB