- #1

Charlotte87

- 21

- 0

## Homework Statement

Let f(x)=x/8 be the density of X on [0,4], zero elsewhere.

a) Show that f(x) is a valid density and compute E(X)

b) Define Y=1/X. Calculate E(Y)

c) Determine the density function for Y

## The Attempt at a Solution

a) is just really basic. I've solved that one.

b) Without any "fuss" about it, i set g(x)=1/x, and use the following formula

[itex]\int^4_0(1/x*x/8)[/itex] = 1/2

c) Here the problem starts... So from my lecture notes i know that

[itex]F_{Y}(y)=F_{X}(g^{-1}(y))[/itex]

Y=1/X --> X=1/Y=g^-1(y)

Using this i can write the above as

[itex]F_{Y}(y)=F_{X}(g^{-1}(1/y))[/itex]

the pdf is then:

[itex]f_{Y}(y)=f_{X}(1/y)*(-1/y^{2})[/itex]

From here, I do not know how to proceed, any clues?