Sunday, January 13, 2019

probability - Let X be a continuous random variable with density function fX. What is Y=aX+b?



Let X be a continuous random variable with density function fX and let a,b>0.



What is Y=aX+b?



I need some help with this one. And I am quite sure it is not afX+b.



Answer



We have that
Pr{aX+by}=Pr{X(yb)/a}=(yb)/afX(x)dx.


Using the substitution x=(tb)/a, we obtain that
Pr{Yy}=(yb)/afX(x)dx=y1afX((tb)/a)dt

and the density function fY(y)=a1fX((ya)/b).




In general, if Y=g(X) with a monotone function g, we have that
fY(y)=|ddy(g1(y))|fX(g1(y)),


where g1 denotes the inverse function (see here for more details). In this particular case g(x)=ax+b for xR.


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f:AB and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...