Maximum Likelihood Estimator for Exponential Density Function

[twoski]

Homework Statement

Given f(x;λ) = $cx^{2}e^{-λx}$ for x ≥ 0

Determine what c must be (as a function of λ) then determine the maximum likelihood estimator of λ.

The Attempt at a Solution

So I'm supposed to integrate this from 0 to infinity, from what i can gather.

Let u = $x^{2}$, du = 2xdx, dv = $e^{-λx}$ and v = $-e^{-λx} / λ$

After a bit of work i end up with:

-c/λ [ $x^{2}e^{-λx}|_{0}^{∞} + 2( xe^{-λx}/λ |^{∞}_{0})$ ]

What throws me off is that evaluating this leaves me with -c/λ( 0 ), which has to be wrong...

 

[Dick]

Homework Statement

Given f(x;λ) = $cx^{2}e^{-λx}$ for x ≥ 0

Determine what c must be (as a function of λ) then determine the maximum likelihood estimator of λ.

The Attempt at a Solution

So I'm supposed to integrate this from 0 to infinity, from what i can gather.

Let u = $x^{2}$, du = 2xdx, dv = $e^{-λx}$ and v = $-e^{-λx} / λ$

After a bit of work i end up with:

-c/λ [ $x^{2}e^{-λx}|_{0}^{∞} + 2( xe^{-λx}/λ |^{∞}_{0})$ ]

What throws me off is that evaluating this leaves me with -c/λ( 0 ), which has to be wrong...

You have to integrate the second term from 0 to infinity, not just evaluate it. You'll need to integrate by parts again.

 

[Ray Vickson]

Homework Statement

Given f(x;λ) = $cx^{2}e^{-λx}$ for x ≥ 0

Determine what c must be (as a function of λ) then determine the maximum likelihood estimator of λ.

The Attempt at a Solution

So I'm supposed to integrate this from 0 to infinity, from what i can gather.

Let u = $x^{2}$, du = 2xdx, dv = $e^{-λx}$ and v = $-e^{-λx} / λ$

After a bit of work i end up with:

-c/λ [ $x^{2}e^{-λx}|_{0}^{∞} + 2( xe^{-λx}/λ |^{∞}_{0})$ ]

What throws me off is that evaluating this leaves me with -c/λ( 0 ), which has to be wrong...

It might be easier to recognize that
$$x e^{-\lambda x} = - \frac{\partial}{\partial \lambda} e^{- \lambda x},$$
and so forth.