(1) Transformation of random variable X(2) Transformation of jointly distributed random variables X and Y(3) Other formulasMath 370X - Lecture 9Transformations of random variablesSaumil PadhyaNovember 7, 2016(1) Transformation of random variable XSuppose X is continuous random variable, and u(x) is a one-to-one function (so an inverse of uexists). Also assume that u is either strictly increasing or decreasing.The random variable Y = u(X) is called a transformation of X.Let v be the inverse function of u. Then the pdf of Y can be found in one of 2 ways -(i)fY(y) = fX(v(y)) · |v0(y)| if u is a strictly increasing function.(ii)Fy(Y ) = P (Y ≤ y) = P (u(X) ≤ y) = P (X ≤ v(y)) = FX(v(y)), and fY(y) = F0Y(y)(2) Transformation of jointly distributed random variables X and YX and Y are jointly distributed. Suppose u and v are functions of x and y withU=u(X, Y) andV = v(X, Y ). We want to find the joint density function of U and V, g(u,v).We start by expressing x and y in terms of u and v: x = h(u, v) and y = k(u, v).We then define the “Jacobian” of the transformation to beJ =∂h∂u·∂k∂v−∂h∂v·∂k∂uThen, we can find the joint density function g(u,v) as followsg(u, v) = f(h(u, v), k(u, v)) · J1(3) Other formulasThe coefficient of correlation between 2 random variables X and Y is defined aspXY=Cov(X, Y )σX· σYThis is different from coefficient of variation of a random variable X which is definedasCV (X)
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