The random variable X is exponential with parameter λ=1 . The random variable Y is defined by Y=g(X)=1/(1+X) .

Question

a) The inverse function h , for which h(g(x))=x , is of the form ay^b+c . Find a , b , and c .

b) For y∈(0,1] , the PDF of Y is of the form fY(y)=y^a*e^((b/y)+c) . Find a , b , and c .

Answer 1

h(x) = 1/x - 1 = (1-x)/x
so, ...

Answer 2

so yes h(x) = 1/x - 1 = (1-x)/x
but what do you do then?
derivative of (1-x)/x = -1/x^2

Answer 3

a) (y^-1)-1 So a=1,b=-1,c=-1

Answer 4

b)y^-2*e^(-1/y+1). so a=-2,b=-1,c=1