1. ## [SOLVED] Density function

I've got this problem:
f(y) = $(1/\alpha)my^{m-1}e^{{-y}^{m}/\alpha}$, y>0
0, elsewhere

Now I'm supposed to find $E(Y^k)$ for any positive integer k.

I start out like this:

$E(Y^k)$ = $\int_0^\infty y^{k}f(y) dy$ = $\int_0^\infty y^{k} (1/\alpha)my^{m-1}e^{{-y}^{m}/\alpha} dy$

I then move out $(1/\alpha)$ and m outside the integral =

$(1/\alpha)m$ $\int_0^\infty y^{k} y^{m-1}e^{{-y}^{m}/\alpha} dy$ = $(1/\alpha)m$ $\int_0^\infty y^{k+m-1}e^{{-y}^{m}/\alpha} dy$

and from here I don't know how to go on. The correct answer is $\Gamma (k/m+1)\alpha ^{k/m}$

and I know that $\Gamma (\alpha) = \int_0^\infty y^{\alpha-1}e^{-y} dy$

but I don't see how $(1/\alpha)m$ $\int_0^\infty y^{k+m-1}e^{{-y}^{m}/\alpha} dy$ can be rewritten to the correct answer. Thanks in advance for your help.

2. Originally Posted by approx
I've got this problem:
f(y) = $(1/\alpha)my^{m-1}e^{{-y}^{m}/\alpha}$, y>0
0, elsewhere

Now I'm supposed to find $E(Y^k)$ for any positive integer k.

I start out like this:

$E(Y^k)$ = $\int_0^\infty y^{k}f(y) dy$ = $\int_0^\infty y^{k} (1/\alpha)my^{m-1}e^{{-y}^{m}/\alpha} dy$

I then move out $(1/\alpha)$ and m outside the integral =

$(1/\alpha)m$ $\int_0^\infty y^{k} y^{m-1}e^{{-y}^{m}/\alpha} dy$ = $(1/\alpha)m$ $\int_0^\infty y^{k+m-1}e^{{-y}^{m}/\alpha} dy$

and from here I don't know how to go on. The correct answer is $\Gamma (k/m+1)\alpha ^{k/m}$

and I know that $\Gamma (\alpha) = \int_0^\infty y^{\alpha-1}e^{-y} dy$

but I don't see how $(1/\alpha)m$ $\int_0^\infty y^{k+m-1}e^{{-y}^{m}/\alpha} dy$ can be rewritten to the correct answer. Thanks in advance for your help.

3. mr fantastic: I'm sorry to say that I still don't understand how to rewrite that expression into the right answer.

4. Originally Posted by approx
mr fantastic: I'm sorry to say that I still don't understand how to rewrite that expression into the right answer.
The first three lines of the reference I've given you are crystal clear I would have thought. What exactly don't you understand?

5. Thanks for your fast answer. I don't understand which substitutions I'm supposed to do. Should I let:

$u = y^m/\alpha$ ?

which gives $(1/\alpha)m$ $\int_0^\infty y^{k+m-1}e^{{-u}} du$ ?

6. Originally Posted by approx
Thanks for your fast answer. I don't understand which substitutions I'm supposed to do. Should I let:

$u = y^m/\alpha$ ?

which gives $(1/\alpha)m$ $\int_0^\infty y^{k+m-1}e^{{-u}} du$ ?
Your substitution is incorrect for a number of reasons:

1. You still have y in the integral, there should only be u's.
2. You have not substituted the correct exprssion for dy. $dy \neq du$ !

Note that $u = \frac{y^m}{\alpha} \Rightarrow dy = \frac{\alpha}{m \, y^{m-1}}\, du$ and $y = (\alpha \, u)^{1/m}$.

It's expected that at this level you can correctly substitute a change of variable in an integral.

7. Ok. So I've done some thinking and came up with this substitution:

$u = y^m$, which gives $y = u^{1/m}$ and $dy = (1/m) u^{{(1/m)}-1} du$. Right?

And then I go on:

$(m/\alpha) \int_0^\infty u^{({1/m})^{k+m-1}}e^{-u/\alpha}(1/m) u^{(1/m)-1}du$

I move $1/m$ outside which gives $(1/m)(m/\alpha) = (1/\alpha)$ outside the integral.

$(1/\alpha) \int_0^\infty u^{({1/m})^{k+m-1}}e^{-u/\alpha}(1/m) u^{(1/m)-1}du$

=

$(1/\alpha) \int_0^\infty u^{(({(k-1)}/m)+1)}e^{-u/\alpha} u^{(1/m)-1}du$

I put together the u:s and receives:

$(1/\alpha) \int_0^\infty u^{k/m}e^{-u/\alpha} du$

Am I right so far?

8. Originally Posted by approx
Ok. So I've done some thinking and came up with this substitution:

$u = y^m$, which gives $y = u^{1/m}$ and $dy = (1/m) u^{{(1/m)}-1} du$. Right?

And then I go on:

$(m/\alpha) \int_0^\infty u^{({1/m})^{k+m-1}}e^{-u/\alpha}(1/m) u^{(1/m)-1}du$

I move $1/m$ outside which gives $(1/m)(m/\alpha) = (1/\alpha)$ outside the integral.

$(1/\alpha) \int_0^\infty u^{({1/m})^{k+m-1}}e^{-u/\alpha}(1/m) u^{(1/m)-1}du$

=

$(1/\alpha) \int_0^\infty u^{(({(k-1)}/m)+1)}e^{-u/\alpha} u^{(1/m)-1}du$

I put together the u:s and receives:

$(1/\alpha) \int_0^\infty u^{k/m}e^{-u/\alpha} du$

Am I right so far?
Yes. Now substitute $w = \frac{u}{\alpha} \Rightarrow u = \alpha w$.

9. Thank you! I got the right answer after the last substitution.