Exploring the Exponential Distribution: Generating Function Method

[sarperb]

Homework Statement

$F_{X}(x)= λe^{-λx} \;for\; x>0 \;\;\;and \;0 \;otherwise$

After finding the characteristic function for the Exponential Distribution, which is (I could do this without problem);
$F_{X}(k)=λ(λ-ik)^{-1}$

Now the question is;

Let $X_1,X_2,\ldots,X_i$ be i.i.d. exponential random variables with parameter λ and let;

$Y_N=\sum_{i=1}^{N}X_i$ (Sum starts from i=1, I am new to LaTeX, I'm not sure if this is the right way to express the end points of the sum)

Using the generating function method, show that the pdf of $Y_N$ is given as;

$f_Y(y) = λ\frac{(λy)^{N-1}}{(N-1)!}e^{-λy}$

Homework Equations

$\Gamma(n) = \int_0^{\infty}t^{n-1}e^{-t}dt$
Which is (n-1)! for n>0 together with $\Gamma(1/2)=\sqrt{\pi}$

Also with a simple substutition of t=az dt = adz
$\Gamma(n) = a^n\int_0^{\infty}z^{n-1}e^{-za}dz$

I used this to show in the same question that
$<X^n>= n!λ^{-n}$

There is a Hint in this part of the question which says (exact copy);
"You can do this without having to explicitly do the k-integral of the
inverse Fourier transform. Instead show that this integral can be written as a
higher-order derivative with respect to a parameter inside a simpler integral,
whose result you already now"

The Attempt at a Solution

From the fact that the sum of random variables applies to generating functions as multipication, it can easily be found that;
$F_{Y}(k)=λ^N(λ-ik)^{-N}$

Now my first problem is taking the inverse Fourier Transform of this guy because I am not sure what the end points of the integral should be. In the first part where I was finding the Characteristic Function of the Exponential Distribution, it was easy to see since $F_X(x)$ was defined to be 0 when x is negative. But now taking the inverse Fourier transform, should I leave the limits from -infinity to infinity as it is for the usual Fourier Transform, or should they be from 0 to infinity?

The answer to this question won't help me solve my problem since I tried with both, but I want to learn how should I be thinking here to get to the right answer.

So the Fourier Transform looks like;

$f_Y(y) = λ^N \int_{0 or -\infty}^{\infty}(λ-ik)^{-N}e^{-iky}dk = I$

The result is given but I can't get myself to it. I played with this for hours and I am at a point where since I did focus on something for too long, I lost perspective and can't have any new ideas to try.

We know I am not to solve the integral explicitly, instead change it into something I already now. I tried;

$(λ-ik)^{-N} =\frac{i^N}{N!}\frac{d^N(λ-ik)^{-1}}{dk^N}$

I played with the Gamma Function etc. By the way I should say, I don't think we are meant to be familiar with the Incomplete Gamma Function.
Since I need to get (N-1)! at the bottom of the fraction I know somehow the reciprocal of the Gamma Function is to be found here, but I don't think I am meant to know the reciprocal of the Gamma Function, so I need to somehow get (N-1)! out and from the remaining integral get y^(N-1).

Also please note that if the initial integral is called I, then;

$Ie^{λy}=λ^N\int_{0 or -\infty}^{\infty}(λ-ik)^{-N}e^{(λ-ik)y}dk$

I of course also tried the substition of u = (λ-ik) du = -idk idu = dk and several other substitions similar to this.

I would have written a lot more about my hours of attempts, but I really don't think spending hours on typing in LaTeX is necessary at this point :)
I would really appreciate if someone could push me in the right direction since after hours I am stuck at the same thoughts and can't continue. :)

P.S. I forgot 1/2pi in the inverse Fourier transforms :)

Thank you for all your help.

 

[mighty2000]

Thank you for your detailed explanation of your attempts at solving this problem. It seems like you have a good understanding of the concepts involved, but are just struggling with the specific steps to take in order to arrive at the correct solution. Here are a few suggestions that may help guide you in the right direction:

1. Consider using integration by parts to simplify the integral. This technique can often be useful in situations where you have an integral involving a product of two functions, as is the case here.

2. Since the hint mentions taking a higher-order derivative with respect to a parameter, it may be helpful to think about how you could rewrite the integral in terms of a different variable. For example, you could try substituting u = λ-ik and see if that leads you to a simpler form.

3. Remember that you are trying to get a result involving (N-1)! at the bottom of the fraction. Think about how you could get this factorial from your integral, and see if there are any techniques or identities involving the Gamma function that could help you achieve this.

Hopefully these suggestions will help you make some progress on this problem. Good luck!