E(X^r) from Weibull Distribution is equal to Gamma Fn

[Nick Jarvis]

Homework Statement

I have f(x) = Bx^B-1e^{-x^B}

I need to show that E(Xr) = Ƭ(Gamma)(R/B + 1)

Homework Equations

I know that E(X^r) = f(x)

The Attempt at a Solution

Attempt at part of solution

I started by saying let u = xB so du = Bx^B-1dx

Can I say then that if u = xB, then X^r = u^r/B

That’s my first question. If I am on the right lines I will integrate. I think I have asked this before, but it got removed as I hadn’t followed the guidelines. Hoping that is set out properly. I am integrating by substitution. Assuming this is possible and I don’t HAVE to integrate by parts.Many thanks
[/B]

 

[Ray Vickson]

Homework Statement

I have f(x) = Bx^B-1e^{-x^B}

I need to show that E(Xr) = Ƭ(Gamma)(R/B + 1)

Homework Equations

I know that E(X^r) = f(x)

The Attempt at a Solution

Attempt at part of solution

I started by saying let u = xB so du = Bx^B-1dx

Can I say then that if u = xB, then X^r = u^r/B

That’s my first question. If I am on the right lines I will integrate. I think I have asked this before, but it got removed as I hadn’t followed the guidelines. Hoping that is set out properly. I am integrating by substitution. Assuming this is possible and I don’t HAVE to integrate by parts.Many thanks[/B]

(1) Please refrain from putting all of your message in a bold font; it looks like you are yelling at us.
(2) When writing you must distinguish clearly between $u = xB$ and $u = x^B$.
(3) And, yes, of course, if $u = x^B$ then $x^r = u^{B/r}$ for any $x > 0$. How could it be otherwise?
(4) I hope you were are being asked to prove that
$$ E X^r = \Gamma \left( \frac{r}{B}+1 \right) \; ?$$
rather than
$$ E X^r = \Gamma \left( \frac{r}{B+1} \right) \: ?$$
because the second one of these is false.

 

[Nick Jarvis]

Thanks. I copied and pasted from Word and assume the bold was inherited from the the 3 titles in the template. Apologies for that.

I cannot work out to insert nice equations like the two that you have inserted above. And yes, I need to prove that:

E(X^r)=Γ((r/B)+1) - your first equation

When I ask 'Can I say then that if u = xB, then X^r = u^r/B' I meant is this the correct way of starting to solve it? Or am I on a hiding to nothing?

Many thanks

 

[Ray Vickson]

Thanks. I copied and pasted from Word and assume the bold was inherited from the the 3 titles in the template. Apologies for that.

I cannot work out to insert nice equations like the two that you have inserted above. And yes, I need to prove that:

E(X^r)=Γ((r/B)+1) - your first equation

When I ask 'Can I say then that if u = xB, then X^r = u^r/B' I meant is this the correct way of starting to solve it? Or am I on a hiding to nothing?

Many thanks

You can get rid of the bold font, just by making sure your input occurs after the "[/B]" delimiter. In this forum, "[B ]" turns on bold and "[/B ]" turns it off. (Note: I inserted extra space after the "B" and before the "]" to prevent the processor from actually switching to bold, but there should be no space between them.)

To insert "nice" equations, just use LaTeX; a stripped-down version of it comes loaded into this Forum. For an in-line equation, use # # d = a + b c^2 \int_0^1 x^3 dx # # (with no space between the two #'s at the start and at the end); that produces $d = a + b c^2 \int_0^1 x^3 \, dx$. For a "dsplayed" equation (on its own, separate line) use two $ signs (with no space between them) at the start and at the end. That gives
$$d = a + b c^2 \int_0^1 x^3 \, dx$$
If you search in this Forum for a "LaTeX tutorial", I am sure you will find one. To see the actual typed commands for a LaTeX expression, just right-click on the expression or equation and ask for a display math as tex.

As to your question: the fall-back position is to always try it yourself, to see what you get. If it works, you are done; if it fails, you need to try something else. But, try it first.