Chapter 4 pp. 153-210

1. Chapter 4pp. 153-210 William J. Pervin The University of Texas at Dallas Richardson, Texas 75083

2. Chapter 3 Pairs of Random Variables

3. Chapter 4 4.1 Joint CDF: The joint CDF FX,Y of RVs X and Y is FX,Y(x,y) = P[X ≤ x, Y ≤ y]

4. Chapter 4 0 ≤ FX,Y(x,y) ≤ 1 If x1 ≤ x2 and y1 ≤ y2 then FX,Y(x1,y1) ≤ FX,Y(x2,y2) FX,Y(∞,∞) = 1

5. Chapter 4 4.2 Joint PMF: The joint PMF of discrete RVs X and Y is PX,Y(x,y) = P[X = x, Y = y] SX,Y = SX x SY

6. 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 PX,Y(x,y)

7. Chapter 4 4.3 Marginal PMF: For discrete RVs X and Y with joint PMF PX,Y(x,y), PX(x) = ΣySY PX,Y(x,y) PY(y) = ΣxSX PX,Y(x,y)

8. Chapter 4 4.4 Joint PDF: The joint PDF of continuous RVs X and Y is function fX,Y such that FX,Y(x,y) = ∫–∞y ∫–∞x fX,Y(u,v)dudv fX,Y(x,y) = ∂2FX,Y(x,y)/∂x∂y

9. Chapter 4 fX,Y(x,y) ≥ 0 for all (x,y) FX,Y(x,y)(∞,∞) = 1

10. Chapter 4 4.5 Marginal PDF: If X and Y are RVs with joint PDF fX,Y, the marginal PDFs are fX(x) = Int{fX,Y(x,y)dy,-∞,-∞} fy(x) = Int{fX,Y(x,y)dx,-∞,-∞}

11. Chapter 4 4.6 Functions of Two RVs: Derived RV W=g(X,Y) X,Y discrete: PW(w) = Sum{PX,Y(x,y)|(x,y):g(x,y)=w}

12. Chapter 4 X,Y continuous: FW(w) = P[W ≤ w] = ∫∫g(x,y)=w fX,Y(x,y)dxdy Example: If W = max(X,Y), then FW(w) = FX,Y(w,w) = ∫y≤w ∫x ≤w fX,Y (x,y)dxdy

13. 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)PX,Y(x,y) Continuous: E[W] = ∫ ∫ g(x,y)fX,Y(x,y)dxdy

14. Chapter 4 Theorem: E[Σgi(X,Y)] = ΣE[gi(X,Y)] In particular: E[X + Y] = E[X] + E[Y]

15. Chapter 4 The covariance of two RVs X and Y is Cov[X,Y] = σXY = E[(X – μX)(Y – μY)] Var[X + Y] = Var[X] + Var[Y] + 2Cov[X,Y]

16. Chapter 4 The correlation of two RVs X and Y is rX,Y = E[XY] Cov[X,Y] = rX,Y – μX μY Cov[X,X] = Var[X] and rX,X = E[X2] Correlation coefficient ρX,Y=Cov[X,Y]/σXσY

17. Chapter 4 4.10 Independent RVs: Discrete: PX,Y(x,y) = PX(x)PY(y) Continuous: fX,Y(x,y) = fX(x)fY(y)

18. Chapter 4 For independent RVs X and Y: E[g(X)h(Y)] = E[g(X)]E[h(Y)] rX,Y = E[XY] = E[X]E[Y] Cov[X,Y] = σX,Y = 0 Var[X + Y] = Var[X] + Var[Y]