Lecture1.TransformationofRandomVariables

1 Lecture 1. Transformation of Random VariablesSuppose we are given a random variableXwith densityfX(x). We apply a functiongto produce a random variableY=g(X). We can think ofXas the input to a blackbox ,andYthe output. We wish to find the density or distribution function the technique for the example in Figure e-x1/2-1f (x)x-axisXYyX-Sqrt[y]Sqrt[y]Y = X2 Figure ,and thenfYby haveFY(y)=0fory<0. Ify 0 ,thenP{Y y}=P{ y x y}.Case y 1 (Figure ). ThenFY(y)=12 y+ y012e xdx=12 y+12(1 e y).1/2-1x-axis-Sqrt[y]Sqrt[y]f (x)XFigure >1 (Figure ). ThenFY(y)=12+ y012e xdx=12+12(1 e y).The density ofYis 0 fory<0 and21/2-1x-axis-Sqrt[y]Sqrt[y]f (x)X1'Figure (y)=14 y(1 +e y),0<y<1;fY(y)=14 ye y,y> Figure for a sketch offYandFY. (You can takefY(y) to be anything you like aty= 1 because{Y=1}has probability zero.)

1. The joint density of two random variables X 1 and X 2 is f(x 1,x 2)=2e−x 1e−x 2, where 0 <x 1 <x 2 <∞;f(x 1,x 2) = 0 elsewhere. Consider the transformation Y 1 =2X 1,Y 2 = X 2 −X 1. Find the joint density of Y 1 and Y 2,and conclude thatY 1 and Y 2 are independent. 2. Repeat Problem 1 with the following new data. The joint density is ...

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  Joint, Density, Lecture1, Transformationofrandomvariables, Joint density

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