Chapter 3 pp. 101-152

1. Chapter 3pp. 101-152 William J. Pervin The University of Texas at Dallas Richardson, Texas 75083

2. Chapter 3 Continuous Random Variables

3. 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)

4. Chapter 3 Definition: A RV X is continuous if its CDF FX is continuous

5. 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

6. Chapter 3 For a continuous RV X with PDF fX(x): • fX(x) ≥ 0 for all x • FX(x) = ∫-∞x fX(u)du • ∫-∞+∞ fX(x)dx = 1 Note: We do not require fX(x) ≤ 1

7. Chapter 3 P[x1 < X ≤ x2] = ∫x1x2 fX(x)dx Note that endpoints don’t matter!

8. 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

9. 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

10. 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

11. 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

12. 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).

13. 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]

14. 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}

15. 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

16. 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)

17. 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)

18. 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

19. Chapter 3 3.7 Probability Models for Derived RVs: If Y = g(X), how to determine fY(y) from g(X) and fX(x): • Find CDF FY(y) = P[Y≤y] • Take derivative fY(y) = dFY(y)dy

20. 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!

21. Chapter 3 3.8 Conditioning a Continuous RV 3.9 MATLAB