Transcription of Lecture1.TransformationofRandomVariables
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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.)yf (y)Y1'yF (y)Y112y+12(1-e-y)12+12(1-e-y)'Figure ,and thenFYby integration; seeFigure We havefY(y)|dy|=fX( y)dx+fX( y)dx; we write|dy|because proba-bilities are never negative.
4. A random variable Xhas density f(x)=ax2 on the interval [0,b]. Find the density of Y= X3. 5. The Cauchydensityis given by f(y)=1/[π(1+y2)] for all real y. Show that one way to produce this density is to take the tangent of a random variable Xthat is uniformly distributed between −π/2 and π/2.
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