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week 91 Example A device containing two key components fails when and only when both components fail. The lifetime, T 1 and T 2, of these components are independent with a common density function given by The cost, X, of operating the device until failure is 2T 1 + T 2. Find the density function of X.

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week 92 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 p X,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 p X (x) and p Y (y).

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week 93 Example Suppose X~ Poisson(λ 1 ) independent of Y~ Poisson(λ 2 ). Find the distribution of X+Y.

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4 Convolution - Continuous case Suppose X, Y random variables with joint density function f X,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 f X (x) and f Y (y).

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week 95 Example X, Y independent each having Exponential distribution with mean 1/λ. Find the density for W=X+Y.

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week 96 Some Recalls on Normal Distribution If Z ~ N(0,1) the density of Z is If X = σZ + μ then X ~ N(μ, σ 2 ) and the density of X is If X ~ N(μ, σ 2 ) then

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week 97 More on Normal Distribution If X, Y independent standard normal random variables, find the density of W=X+Y.

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week 98 In general, If X 1, X 2,…, X n i.i.d N(0,1) then X 1 + X 2 +…+ X n ~ N(0,n). If,,…, then If X 1, X 2,…, X n i.i.d N(μ, σ 2 ) then S n = X 1 + X 2 +…+ X n ~ N(nμ, nσ 2 ) and

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9 Sum of Independent χ 2 (1) random variables Recall: The Chi-Square density with 1 degree of freedom is the Gamma(½, ½) density. If X 1, X 2 i.i.d with distribution χ 2 (1). Find the density of Y = X 1 + X 2. In general, if X 1, X 2,…, X n ~ χ 2 (1) independent then X 1 + X 2 +…+ X n ~ χ 2 (n) = Gamma(n/2, ½). Recall: The Chi-Square density with parameter n is

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week 910 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.

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week 911 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 f U,V (u,v) from f X,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: R 2 R 2 defined by u = f(x,y), v = g(x,y).

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week 912 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.

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week 913 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

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week 914 Example Evaluate where Axy is bounded by y = x, y = ex, xy = 2 and xy = 3.

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week 915 Change-of-Variable for Joint Distributions Theorem Let X and Y be jointly continuous random variables with joint density function f X,Y (x,y) and let D XY = {(x,y): f X,Y (x,y) >0}. If the mapping T given by T(x,y) = (u(x,y),v(x,y)) maps D XY onto D UV. 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 D UV.

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week 916 Example Let X, Y have joint density function given by Find the density function of

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week 917 Example Show that the integral over the Standard Normal distribution is 1.

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week 918 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

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week 919 Example Suppose X, Y are independent N(0,1). The density of is

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week 920 Example – F distribution Suppose X ~ χ 2 (n) independent of Y ~ χ 2 (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).

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week 921 Example – t distribution Suppose Z ~ N(0,1) independent of X ~ χ 2 (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)

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week 922 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

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week 923 Derivation of Beta Distribution Let X 1, X 2 be independent χ 2 (1) random variables. We want the density of Can define the following transformation

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Special random variables Chapter 5 Some discrete or continuous probability distributions.

Special random variables Chapter 5 Some discrete or continuous probability distributions.

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