From the results of part a find the density of Y 1 X X is a

From the results of part a)., find the density of Y =1/ X .

X is a continuous random variable with density fX (x) . a). A new random variable Y is defined by Y = g(X) =1/ X . Find the density fY (y) in terms of fX (x) . b). Now, suppose X is a Cauchy random variable with parameter a > 0 . Thus its density is fX (x) = (a / infinity From the results of part a)., find the density of Y =1/ X . infinity

Solution

Let f(x) be pdf of x.

Then Y = 1/x is defined for all values of x except 0

f_y(Y) = f() , for y not equals 0

b) If f(x) =

Substitute x =1./y

fy(y) = a(1+a²y²)/piy², for all y except at 0