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MULTIPLE CONTINUOUS r.v

This document explores the fundamentals of joint continuous random variables (r.v.) X and Y, emphasizing the joint cumulative distribution function (CDF) and joint probability density function (PDF). We examine the marginal PDFs of X and Y, the expectations of functions of X and Y, including covariance and correlation coefficients. Additionally, we discuss the implications of independence between the r.v.'s and the characteristics of jointly Gaussian random variables. Key chapters outline the methods for determining expectations and conditional PDFs essential for understanding statistical relationships.

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MULTIPLE CONTINUOUS r.v

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  1. MULTIPLE CONTINUOUS r.v • JCFD (joint CDF) for r.v X & Y: FX,Y(x,y) = P[X < x, Y < y] • JPDF : Chpt. 5

  2. Marginal pdf • X, Y r.v’s with JPDF fX,Y(x,y). • the marginal pdf of X is just fX(x) : • Similarly for the marginal pdf of Y: Chpt. 5

  3. Functions of 2 r.v’s • 2 r.v’s X, Y with JPDF fX,Y(x,y). We may be interested in some function W=g(X,Y) • To determine the corresponding pdf and cdf of W : fW(w), and FW(w). Chpt. 5

  4. Expectation Values for 2 r.v’s X & Ywith respect to JPDF fX,Y(x,y) • E[W]=E[g(X,Y)] - w.r.t JPDF fX,Y(x,y) • E[X+Y] = E[X] +E[Y] • COVARIANCE • CORRELATION COEFFICIENT Chpt. 5

  5. CONDITIONAL PDF • conditional expectation value of g(X,Y) given y : will use conditional pdf fX|Y(x|y) Chpt. 5

  6. INDEPENDENT R.V’s • X, Y are independent • X, Y indep. then Cov[X,Y] = 0 • E[ g(X). h(Y) ] = E[g(X)] . E[h(Y)] • Var[X+Y] = Var[X] + Var[Y] Chpt. 5

  7. JOINTLY GAUSSIAN R.V’s • Jointly Gaussian : if all the marginal PDF’s of the r.v’s are Gaussian • bivariate Gaussian PDF (2 r.v’s) : • marginal PDF’s fX(x), fY(y) • conditional PDF of Y given X • correlation coefficient rX,Y Chpt. 5

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