1. The Erik Jonsson School of Engineering and Computer Science Chapter 4 pp. 153-210 William J. Pervin The University of Texas at Dallas Richardson, Texas 75083

2. The Erik Jonsson School of Engineering and Computer Science Chapter 3 Pairs of Random Variables

3. The Erik Jonsson School of Engineering and Computer Science Chapter 4 4.1 Joint CDF: The joint CDF F X,Y of RVs X and Y is F X,Y (x,y) = P[X ≤ x, Y ≤ y]

4. The Erik Jonsson School of Engineering and Computer Science Chapter 4 0 ≤ F X,Y (x,y) ≤ 1 If x 1 ≤ x 2 and y 1 ≤ y 2 then F X,Y (x 1,y 1 ) ≤ F X,Y (x 2,y 2 ) F X,Y (∞,∞) = 1

5. The Erik Jonsson School of Engineering and Computer Science Chapter 4 4.2 Joint PMF: The joint PMF of discrete RVs X and Y is P X,Y (x,y) = P[X = x, Y = y] S X,Y = S X x S Y

6. The Erik Jonsson School of Engineering and Computer Science Chapter 4 For discrete RVs X and Y and any B  X x Y, the probability of the event {(X,Y)  B} is P[B] = Σ (x,y)  B P X,Y (x,y)

7. The Erik Jonsson School of Engineering and Computer Science Chapter 4 4.3 Marginal PMF: For discrete RVs X and Y with joint PMF P X,Y (x,y), P X (x) = Σ y  S Y P X,Y (x,y) P Y (y) = Σ x  S X P X,Y (x,y)

8. The Erik Jonsson School of Engineering and Computer Science Chapter 4 4.4 Joint PDF: The joint PDF of continuous RVs X and Y is function f X,Y such that F X,Y (x,y) = ∫ –∞ y ∫ –∞ x f X,Y (u,v)dudv f X,Y (x,y) = ∂ 2 F X,Y (x,y)/∂x∂y

9. The Erik Jonsson School of Engineering and Computer Science Chapter 4 f X,Y (x,y) ≥ 0 for all (x,y) F X,Y (x,y)(∞,∞) = 1

10. The Erik Jonsson School of Engineering and Computer Science Chapter 4 4.5 Marginal PDF: If X and Y are RVs with joint PDF f X,Y, the marginal PDFs are f X (x) = Int{f X,Y (x,y)dy,-∞,-∞} f y (x) = Int{f X,Y (x,y)dx,-∞,-∞}

11. The Erik Jonsson School of Engineering and Computer Science Chapter 4 4.6 Functions of Two RVs: Derived RV W=g(X,Y) X,Y discrete: P W (w) = Sum{P X,Y (x,y)|(x,y):g(x,y)=w}

12. The Erik Jonsson School of Engineering and Computer Science Chapter 4 X,Y continuous: F W (w) = P[W ≤ w] = ∫∫ g(x,y)=w f X,Y (x,y)dxdy Example: If W = max(X,Y), then F W (w) = F X,Y (w,w) = ∫ y≤w ∫ x ≤w f X,Y (x,y)dxdy

13. The Erik Jonsson School of Engineering and Computer Science Chapter 4 4.7 Expected Values: For RVs X and Y, if W = g(X,Y) then Discrete: E[W] = Σ Σ g(x,y)P X,Y (x,y) Continuous: E[W] = ∫ ∫ g(x,y)f X,Y (x,y)dxdy

14. The Erik Jonsson School of Engineering and Computer Science Chapter 4 Theorem: E[Σg i (X,Y)] = ΣE[g i (X,Y)] In particular: E[X + Y] = E[X] + E[Y]

15. The Erik Jonsson School of Engineering and Computer Science Chapter 4 The covariance of two RVs X and Y is Cov[X,Y] = σ XY = E[(X – μ X )(Y – μ Y )] Var[X + Y] = Var[X] + Var[Y] + 2Cov[X,Y]

16. The Erik Jonsson School of Engineering and Computer Science Chapter 4 The correlation of two RVs X and Y is r X,Y = E[XY] Cov[X,Y] = r X,Y – μX μY Cov[X,X] = Var[X] and r X,X = E[X 2 ] Correlation coefficient ρ X,Y =Cov[X,Y]/σ X σ Y

17. The Erik Jonsson School of Engineering and Computer Science Chapter 4 4.10 Independent RVs: Discrete: P X,Y (x,y) = P X (x)P Y (y) Continuous: f X,Y (x,y) = f X (x)f Y (y)

18. The Erik Jonsson School of Engineering and Computer Science Chapter 4 For independent RVs X and Y: E[g(X)h(Y)] = E[g(X)]E[h(Y)] r X,Y = E[XY] = E[X]E[Y] Cov[X,Y] = σ X,Y = 0 Var[X + Y] = Var[X] + Var[Y]