1. 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.
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2. 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).
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3. Example Suppose X~ Poisson(λ1) independent of Y~ Poisson(λ2). Find the
distribution of X+Y.
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4. 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).
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5. Example
X, Y independent each having Exponential distribution with mean 1/λ. Find the density for W=X+Y.
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6. 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
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7. More on Normal Distribution
If X, Y independent standard normal random variables, find the density of W=X+Y.
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8. 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
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9. 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
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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.
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11. 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).
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12. 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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13. 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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14. Example
Evaluate where Axy is bounded by y = x, y = ex, xy = 2 and xy = 3.
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15. 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 .
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16. Example Let X, Y have joint density function given by
Find the density function of
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17. Example
Show that the integral over the Standard Normal distribution is 1.
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18. 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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19. Example
Suppose X, Y are independent N(0,1). The density of is
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20. 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).
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21. 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)
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22. 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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23. Derivation of Beta Distribution
Let X1, X2 be independent χ2(1) random variables. We want the density of
Can define the following transformation
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