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LFC
08-02-2002, 03:37 PM
okay ppl...help me here:

integrate {exp(-2x)X^y}y! dx

i know from C1 there's a simple relationship here but i cant put my finger on it!

Gandalf
08-02-2002, 03:49 PM
Why do you think it is a simple expression?

The larger y gets, the messier the integral becomes.

The form will be [exp(-2x)][Polynomial of degree y in x, with all coefficients non-zero].

Alya
08-02-2002, 04:12 PM
okay ppl...help me here:

integrate {exp(-2x)X^y}y! dx

i know from C1 there's a simple relationship here but i cant put my finger on it!

assuming you want to integrate from 0 to inf and y is an integer:
change this to look more like Gamma density and you will get (y!)^2

MathGuy
08-02-2002, 04:14 PM
Actually, this looks a lot like a Gamma function problem, assuming the limits of integration are 0 and infinity:

Int (y! * x^y * e^(-2x), dx)
= y! * Int (x^y * e^(-2x), dx)

Substitution:

u = 2x ==> x = u/2 ==> dx = du/2

And now we have:

= y! * Int ((u/2)^y * e^u, du/2)
= y!/2 * Int (2^-y * u^y * e^u, du)
= y! * 2^-(y+1) * Int (u^y * e^u, du)
= y! * 2^-(y+1) * Gamma (y - 1)

But, Gamma (y - 1) = y!

= (y!)^2 * 2^-(y+1).

Alya
08-02-2002, 04:22 PM
(y!)^2 * 2^-(y+1).
This is better then my (y!)^2 :oops:
I guess i should not try to do these w/o paper :)

LFC, which course are you taking?

LFC
08-02-2002, 04:39 PM
oh god...guys (and alya)..i'm sorry i messed up..the y! is a denominator.

so its integrate (1/y!)exp(-2x)X^y dx with limits 0 and inf.

NOW, its supposed to be easy but i cant remember how to arrive at the final answer...which is:

(1/2)^(y+1) a geometric with parameter (and mean) 1.

some kind of a relationship i cant remember..
:-?

alya, i'm doing C3.

slagheap
08-02-2002, 04:47 PM
For C3 you shouldn't have to resort to integrating things (not often anyways). Recognize that this function is proportional to a Gamma density with parameters (y+1, 1/2). Since the Gamma integrates to unity, your integral is y! * (1/2)^(Y+1). The y! cancels and you get your answer.

Alya
08-02-2002, 05:35 PM
For C3 you shouldn't have to resort to integrating things (not often anyways).

And you even don't need to remember exactly what Gamma density is since it is given to you on the exam. All you need to know is that
1. you have seen this function before (so look at the list of available densities).
2. It is very unlikely that you will need to integrate anything by parts (so look at the list of available densities some more).

08-02-2002, 08:21 PM
Why do you have a big X and a little x? Make up your mind or give us more detail!

LFC
08-03-2002, 12:31 PM
Why do you have a big X and a little x? Make up your mind or give us more detail!

the x's are the same..i hope that didnt stump u steve!
well this question was a sub-problem to a bigger problem..i know i wont need to integrate, but u know its just one of those "build-up" questions to the actual exam...it's to help me get more comfortable with gamma and such distributions.

regarding the tables in the C3 exam..is it in booklet form? and in that exact order shown on the web?

MathGuy
08-05-2002, 08:30 AM
Yes, the distributions are given exactly as they are on the website. And, Re:Integrals, I do everything on exams 3 and 4 with integration. Why would I do such a thing? Because I already know it, and I don't feel like learning about all the different distributions. Calculus can be used to solve any ad hoc Loss Distributions question. I gotta be me.

Alya
08-05-2002, 12:47 PM
Re:Integrals, I do everything on exams 3 and 4 with integration. [...] Calculus can be used to solve any ad hoc Loss Distributions question.

I also did this for a while - until i checked the clock. Don't know about you, but it took me about three times as long to integrate something 3 times by parts instead of using E(X^3) for exponential variable or something like this. I just felt that i will be better off spending time on the exam on thinking about problems that i don't know how to do, than doing calculations. (And in case of this particular problem what would you do w/o gamma distribution? Do induction on y? )

Of cause calculus can be used, the question is weather it should be used.

MathGuy
08-05-2002, 03:02 PM
I admit that, as in my solution above, I do use the Gamma function as a solution to an integral I've set up. However, the reason that I use integration to almost any loss distribution problem involving expected values is study time, as in, I'd prefer not to study. I don't have a real "go-getter" attitude about these exams. I'm more of a "I have a set of skills X, while the exam requires set of skills Y. I hope there is a big nough intersection between X and Y so I pass this time." This way, I get to watch lots of TV. Like I said, I gotta be me.

Alya
08-05-2002, 04:43 PM
[...] I'd prefer not to study. [...] This way, I get to watch lots of TV.

:D