1. ## cdf question

I'm having a lot of trouble figuring out how to go about answering this question...

I think part b) is just, Fx = integral of fx(x), and you'll get an expression for Fx...

part a) would be something like..
Fm = P(m <= x)
= 1 - P(m > x)
= 1 - Fx
but I'm not sure how Fx and Fm relate together..

As you can see, I'm pretty confused, so any help in starting off and understanding would be a great help!

2. Hello,

part a) would be something like..
Fm = P(m <= x)
= 1 - P(m > x)
= 1 - Fx
No !

It is said that if M>x, then $\displaystyle \forall i$, $\displaystyle X_i>x$
So :
$\displaystyle \mathbb{P}(M>x)=\mathbb{P}(X_1>x ,X_2>x,\dots,X_n>x)$
Since the Xi's are independent, you can write this as a product :
$\displaystyle \mathbb{P}(M>x)=\mathbb{P}(X_1>x)\dots\mathbb{P}(X _n>x)$
Since they follow the same distribution, this is :
$\displaystyle \mathbb{P}(M>x)=[\mathbb{P}(X_1>x)]^n$
But $\displaystyle \mathbb{P}(X_1>x)=1-\mathbb{P}(X_1 \leq x)=1-F_X(x)$

Hence we get :
$\displaystyle F_M(x)=1-[1-F_X(x)]^n$

I think part b) is just, Fx = integral of fx(x), and you'll get an expression for Fx..
True ! But it's a bit more complicated for the details (not much, don't worry)

And be careful, it's sometimes confusing if the dummy variable of the integral has the same name as one of the boundaries.

$\displaystyle F_X(x)=\int_{-\infty}^x f_X(t) ~dt$
Note that the density function is only defined for t>0.
So if x<0, the integral is 0.
If x>0, the integral is $\displaystyle \int_0^x f_X(t) ~dt$
Can you understand why ?

(don't forget to substitute $\displaystyle \lambda$ with the value you're given)

3. Thanks soo much Moo.

Part a) makes so much sense now that you've stepped it out for me.

For part c), i think you just need to substitute Fx from b) into Fx from the expression from a) to give Ft.
$\displaystyle F_T = 1 - [1 + exp(-\lambda x)[\lambda x + 1]]^n$
Is that the right method?

For part d) I'm a bit lost on what they mean by let $\displaystyle Y = root(n)T$..
Is Fy just Ft from part c) with $\displaystyle Y = root(n)T$? How do you use it?

Thanks in advanced! Just pretty confused at all this..

For c) :
For part c), i think you just need to substitute Fx from b) into Fx from the expression from a) to give Ft.
$\displaystyle F_T = 1 - [1 + exp(-\lambda x)[\lambda x + 1]]^n$
Is that the right method?
Right method ! (because it is said that it goes out as soon as one light fails, so basically, the life time will be the min between the different lights' life time)
But wrong formula..
I get $\displaystyle F_X(x)=1-e^{-\lambda x}(\lambda x+1)$ (for x>0)
It seems that you didn't get the 1... If you have a problem in spotting where your mistake is (or if you think you're correct), provide what you did, and I'll help you go through it. But I'm confident you'll find it
You can "check" it with the fact that the pdf is continuous -> the cdf is continuous -> $\displaystyle F_X(0)$ should be equal to 0 (according to the previous question)

And $\displaystyle F_T$ would then have a more friendly expression ^^

For d) :
$\displaystyle F_Y(y)=\mathbb{P}(Y \leq y)=\mathbb{P}(\sqrt{n} T \leq y)=\mathbb{P}\left(T \leq \frac{y}{\sqrt{n}}\right)$

So what does it look like ? Can you relate it to $\displaystyle F_T(t)=\mathbb{P}(T\leq t)$ ?

For e) :
I don't like it... If it has to be related to d), it means that you have to consider 50 as a large number (since you had the limit as n goes to infinity)...
As an answer to this question, you can "see" that the limit of $\displaystyle F_Y(y)$ looks like the cdf of an exponential distribution. But for the moment, I don't see how to get E(T)