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 The cost, X, of operating the device until failure is 2T1 + T2. Find the density function of X. week 10

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 If X, Y independent then This is known as the convolution of pX(x) and pY(y). week 10

Example Suppose X~ Poisson(λ1) independent of Y~ Poisson(λ2). Find the distribution of X+Y. week 10

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: If is continuous at z then the density function of Z is given by If X, Y independent then This is known as the convolution of fX(x) and fY(y). week 10

Example X, Y independent each having Exponential distribution with mean 1/λ. Find the density for W=X+Y. week 10

Some Recalls on Normal Distribution If Z ~ N(0,1) the density of Z is If X = σZ + μ then X ~ N(μ, σ2) and the density of X is If X ~ N(μ, σ2) then week 10

More on Normal Distribution If X, Y independent standard normal random variables, find the density of W=X+Y. week 10

If X1, X2,…, Xn i.i.d N(0,1) then X1+ X2+…+ Xn ~ N(0,n). If , ,…, then In general, If X1, X2,…, Xn i.i.d N(0,1) then X1+ X2+…+ Xn ~ N(0,n). If , ,…, then If X1, X2,…, Xn i.i.d N(μ, σ2) then Sn = X1+ X2+…+ Xn ~ N(nμ, nσ2) and week 10

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 week 10

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 . Want to find the density of Z. week 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 . 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 . Further, suppose T is a transformation with T(x,y) = (f(x,y),g(x,y)) = (u,v). Then, 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 10

Jacobian Definition: The Jacobian Matrix of the transformation T is given by 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 10

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 where week 10

Example Evaluate where Axy is bounded by y = x, y = ex, xy = 2 and xy = 3. week 10

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 where J(u,v) is the Jacobian of T-1 given by assuming derivatives exists and are continuous at all points in DUV . week 10

Example Let X, Y have joint density function given by Find the density function of week 10

Example Show that the integral over the Standard Normal distribution is 1. week 10

Density of Quotient Suppose X, Y are independent continuous random variables and we are interested in the density of Can define the following transformation . The inverse transformation is x = w, y = wz. The Jacobian of the inverse transformation is given by Apply 2-D change-of-variable theorem for densities to get The density for Z is then given by week 10

Example Suppose X, Y are independent N(0,1). The density of is week 10

Example – F distribution Suppose X ~ χ2(n) independent of Y ~ χ2(m). Find the density of 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 10

Example – t distribution Suppose Z ~ N(0,1) independent of X ~ χ2(n). Find the density of This is the Density for a random variable with a t-distribution with parameter n (often called degrees of freedom). T ~ t(n) week 10

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 If α = β = 1, then X ~ Uniform(0,1). If α = β = ½ , then the density of X is Depending on the values of α and β, density can look like: If X ~ Beta(α,β) then and week 10

Derivation of Beta Distribution Let X1, X2 be independent χ2(1) random variables. We want the density of Can define the following transformation week 10