Transcription of Lecture1.TransformationofRandomVariables
{{id}} {{{paragraph}}}
Example: dental hygienist
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.
7 2.3ATypicalApplication Let Xand Ybe independent,positive random variables with densitiesf X and f Y,and let Z= XY.We find the density of Zby introducing a new random variable W,as follows: Z= XY, W= Y (W= Xwould be equally good).The transformation is one-to-one because we can solve for X,Yin terms of Z,Wby X= Z/W,Y= W.In a problem of this type,we must always
Loading..
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: