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 $\displaystyle p(N) = ( {5} / {6})^{N-1} ({1} /{6})$

And it gives the example of p(N=1,3,5,7,...) = $\displaystyle ({6}/{11})$

Edit:

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

$\displaystyle p(N) = \sum_{x=1}^{\infty} ({5} / {6})^{N-1} ({1} /{6}) = {1}$

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:

$\displaystyle f(x) = \frac{x}{2}$

$\displaystyle E(\frac{1}{x}) = \int_{-\infty}^{\infty} (\frac{1}{x}) f(x) dx = \int_{0}^{2} (\frac{1}{x}) (\frac{x}{2}) dx = {1}$

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 $\displaystyle \leq$ x < 1}

{1 if x $\displaystyle \geq$ 1}

pdf = (i have no idea)