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Example • A device containing two key components fails when and only when both components fail. The lifetime, T1 and T2, of these components are independent with a common density function given by e t fT t 0 t 0 otherwise • The cost, X, of operating the device until failure is 2T1 + T2. Find the density function of X. week 9 1 Convolution • Suppose X, Y jointly distributed random variables. We want to find the probability / density function of Z=X+Y. • Discrete case X, Y have joint probability function pX,Y(x,y). Z = z whenever X = x and Y = z – x. So the probability that Z = z is the sum over all x of these joint probabilities. That is pZ z p X ,Y x, z x . x • If X, Y independent then pZ z p X x pY z x . x This is known as the convolution of pX(x) and pY(y). week 9 2 Example • Suppose X~ Poisson(λ1) independent of Y~ Poisson(λ2). Find the distribution of X+Y. week 9 3 Convolution - Continuous case • Suppose X, Y random variables with joint density function fX,Y(x,y). We want to find the density function of Z=X+Y. Can find distribution function of Z and differentiate. How? The Cdf of Z can be found as follows: FZ z P X Y z zx f x, y dydx X ,Y x y z f x, v x dvdx X ,Y x v z f x, v x dxdv. X ,Y v x If f x, v x dx XY is continuous at z then the density function of Z is given by x • If X, Y independent then f Z z f Z z f x, z xdx XY x f x f z xdx X Y x This is known as the convolution of fX(x) and fY(y). 4 Example • X, Y independent each having Exponential distribution with mean 1/λ. Find the density for W=X+Y. week 9 5 Some Recalls on Normal Distribution • If Z ~ N(0,1) the density of Z is z 1 Z z e 2 2 2 , z • If X = σZ + μ then X ~ N(μ, σ2) and the density of X is x 1 2 f X x e 2 2 2 2 , x • If X ~ N(μ, σ2) then Z X ~ N 0,1. week 9 6 More on Normal Distribution • If X, Y independent standard normal random variables, find the density of W=X+Y. week 9 7 In general, • If X1, X2,…, Xn i.i.d N(0,1) then X1+ X2+…+ Xn ~ N(0,n). • If X 1 ~ N 1 , 12 , X 2 ~ N 2 , 22 ,…, X n ~ N n , n2 then X 1 X 2 X n ~ N 1 n , 12 n2 . • If X1, X2,…, Xn i.i.d N(μ, σ2) then Sn = X1+ X2+…+ Xn ~ N(nμ, nσ2) 2 Sn and X n . ~ N , n n week 9 8 Sum of Independent χ2(1) random variables • Recall: The Chi-Square density with 1 degree of freedom is the Gamma(½ , ½) density. • If X1, X2 i.i.d with distribution χ2(1). Find the density of Y = X1+ X2. • In general, if X1, X2,…, Xn ~ χ2(1) independent then X1+ X2+…+ Xn ~ χ2(n) = Gamma(n/2, ½). • Recall: The Chi-Square density with parameter n is f X x n/2 1 1 n 2 e 2 x x 2 1 n 2 0 0 x otherwise 9 Cauchy Distribution • The standard Cauchy distribution can be expressed as the ration of two Standard Normal random variables. • Suppose X, Y are independent Standard Normal random variables. Let Z Y . Want to find the density of Z. X week 9 10 Change-of-Variables for Double Integrals • Consider the transformation , u = f(x,y), v = g(x,y) and suppose we are interested in evaluating F x, y dAxy . Dxy • Why change variables? In calculus: - to simplify the integrand. - to simplify the region of integration. In probability, want the density of a new random variable which is a function of other random variables. • Example: Suppose we are interested in finding P A f X ,Y x, y dxdy . . A Further, suppose T is a transformation with T(x,y) = (f(x,y),g(x,y)) = (u,v). Then, P A f U ,V u, v dudv. T A • Question: how to get fU,V(u,v) from fX,Y(x,y) ? • In order to derive the change-of-variable formula for double integral, we need the formula which describe how areas are related under the transformation T: R2 R2 defined by u = f(x,y), v = g(x,y). week 9 11 Jacobian • Definition: The Jacobian Matrix of the transformation T is given by f x J T x, y g x f y u , v g x, y y • The Jacobian of a transformation T is the determinant of the Jacobian matrix. • In words: the Jacobian of a transformation T describes the extent to which T increases or decreases area. week 9 12 Change-of-Variable Theorem in 2-dimentions • Let x = f(u,v) and y = g(u,v) be a 1-1 mapping of the region Auv onto Axy with f, g having continuous partials derivatives and det(J(u,v)) ≠ 0 on Auv. If F(x,y) is continuous on Axy then F x, y dxdy F f u, v , g u, v J u, v dudv Axy where x J u, v u y u Auv x v 1 y J x, y v week 9 13 Example • Evaluate xydxdy where Axy is bounded by y = x, y = ex, xy = 2 and xy = 3. Axy week 9 14 Change-of-Variable for Joint Distributions • Theorem Let X and Y be jointly continuous random variables with joint density function fX,Y(x,y) and let DXY = {(x,y): fX,Y(x,y) >0}. If the mapping T given by T(x,y) = (u(x,y),v(x,y)) maps DXY onto DUV. Then U, V are jointly continuous random variable with joint density function given by f xu , v , y u , v J u , v fU ,V u , v X ,Y 0 if u, v DU ,V otherwise where J(u,v) is the Jacobian of T-1 given by x J u, v u y u x v y v assuming derivatives exists and are continuous at all points in DUV . week 9 15 Example • Let X, Y have joint density function given by e x y f X ,Y x, y 0 Find the density function of U if x, y 0 otherwise X . X Y week 9 16 Example • Show that the integral over the Standard Normal distribution is 1. week 9 17 Density of Quotient • Suppose X, Y are independent continuous random variables and we are Y interested in the density of Z . X y • Can define the following transformation z , w x . x • The inverse transformation is x = w, y = wz. The Jacobian of the inverse transformation is given by x J w, z w y w x z 1 0 w y z w z • Apply 2-D change-of-variable theorem for densities to get fW ,Z w, z f X ,Y w, wz w f X w fY wz w • The density for Z is then given by f Z z f X w fY wz w dw week 9 18 Example • Suppose X, Y are independent N(0,1). The density of Z week 9 Y is X 19 Example – F distribution X /n . • Suppose X ~ (n) independent of Y ~ (m). Find the density of Z Y /m χ2 χ2 • This is the Density for a random variable with an F-distribution with parameters n and m (often called degrees of freedom). Z ~ F(n,m). week 9 20 Example – t distribution • Suppose Z ~ N(0,1) independent of X ~ χ2(n). Find the density of T Z X . n • This is the Density for a random variable with a t-distribution with parameter n (often called degrees of freedom). T ~ t(n) week 9 21 Some Recalls on Beta Distribution • If X has Beta(α,β) distribution where α > 0 and β > 0 are positive parameters the density function of X is 1 1 x 1 x 0 x 1 f X x 0 otherwise • If α = β = 1, then X ~ Uniform(0,1). • If α = β = ½ , then the density of X is 1 f X x x1 x 0 for 0 x 1 otherwise • Depending on the values of α and β, density can look like: • If X ~ Beta(α,β) then E X and V X week 9 . 2 1 22 Derivation of Beta Distribution • Let X1, X2 be independent χ2(1) random variables. We want the density of X1 X1 X 2 • Can define the following transformation Y1 X1 X1 X 2 , Y2 X 1 X 2 week 9 23