Chapter 4 pp 153 210 William J Pervin

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

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

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] The Erik Jonsson School of Engineering and Computer Science

Chapter 4 0 ≤ FX, Y(x, y) ≤ 1 If x 1 ≤ x 2 and y 1 ≤ y 2 then FX, Y(x 1, y 1) ≤ FX, Y(x 2, y 2) FX, Y(∞, ∞) = 1 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 PX, Y(x, y) = P[X = x, Y = y] SX, Y = SX x SY 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 PX, Y(x, y) The Erik Jonsson School of Engineering and Computer Science

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) 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 y x FX, Y(x, y) = ∫–∞ f. X, Y(u, v)dudv f. X, Y(x, y) = ∂2 FX, Y(x, y)/∂x∂y The Erik Jonsson School of Engineering and Computer Science

Chapter 4 f. X, Y(x, y) ≥ 0 for all (x, y) FX, Y(x, y)(∞, ∞) = 1 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, -∞} fy(x) = Int{f. X, Y(x, y)dx, -∞} 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: PW(w) = Sum{PX, Y(x, y)|(x, y): g(x, y)=w} The Erik Jonsson School of Engineering and Computer Science

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

Chapter 4 Theorem: E[Σgi(X, Y)] = ΣE[gi(X, Y)] In particular: E[X + Y] = E[X] + E[Y] The Erik Jonsson School of Engineering and Computer Science

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] + 2 Cov[X, Y] 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 ρX, Y=Cov[X, Y]/σXσY The Erik Jonsson School of Engineering and Computer Science

Chapter 4 4. 10 Independent RVs: Discrete: PX, Y(x, y) = PX(x)PY(y) Continuous: f. X, Y(x, y) = f. X(x)f. Y(y) 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] = σX, Y = 0 Var[X + Y] = Var[X] + Var[Y] The Erik Jonsson School of Engineering and Computer Science