Hi there, I am taking a Statistics course at University, but it's been a while since I did any of this stuff so I'm a bit rusty. Here's a question I'm trying to do:

Well for part (i) I simply integrated 2e^{-2x} to get -e^{-2x} for when x is between 0 and infinity. But then I remembered, isn't 0 <= F(x) <= 1 a property of the CDF? And in this case, if x is for example 3, then F(3) = -e^{-6} which certainly isn't between 0 and 1. So where have I wrong here?Let X denote a random variable with probability density function

$\displaystyle f(x) = 2e^{-2x} when 0<x<inf, and = 0 elsewhere$

(i) Derive the CDF of X

(ii) Calculate $\displaystyle E[X^2 + 3X]$

(iii) Let Y = min(V,W), where V and W are independent and each have the same distribution as X. Find the CDF of Y and hence the PDF of Y.

I then thought I could do part (ii) without successfully having done part (i). I used the fact that E[X^2 + 3X] = E[X^2] + 3E[X] and ended up getting an answer of 2, can anyone verify this for me please? Just so I know whether I'm doing things right!

And for part (iii) I decided I needed to know about part (i) first...once I get part (i), I think I might be able to do it, I have something in my notes and 2 random variables being "independent and identically distributed" which I think may apply here.

Many thanks for anyone who can help me out, my exam is on Friday and I need to sort out a few kinks!

Best,

Ruaraidh