Hi i was working through my Mathematical Statistics book and i came across a couple of examples that have got me stumped.

A Die is cast independently N times until a six appears on the up side of the die.

a) Find probability p(N) and prove it is a discrete distribution

b) Find p(N=2,4,6,8,...)

c) Find p(N=3,6,9,12,...)

d) Find the cdf for random variable N

I know that the pmf is

And it gives the example of p(N=1,3,5,7,...) =

Edit:

For a) is this proof that it is a discrete distribution?

Second Question:

Let probability density f(x)=x/2 for 0<x<2 and 0 elsewhere. Compute E(1/X), compute cumulative distribution function and probability density function of Y=1/X, find the expected value and variance of Y.

Heres what i did so far:

As for Y=1/X i have no idea how to get it. Is it a transformation? My book is terrible and i am lost.

cdf =

{0 if x < 0}

{x if 0 x < 1}

{1 if x 1}

pdf = (i have no idea)