New guy "stumped" on a HW problem

Hello, I found this forum and hoped I would be able to get help with a problem that is just consuming all of my time.

It involves finding the Maximum Likelihood Estimator. I apologize in advance for maybe not expressing any formulas correctly on here...I've never typed these in a post like this before so I don't know how to correctly type symbols for "sum" and "product" etc.

Basically, the problem is this:

Density function f(x) = (3/lambda) * x^2 * e^((-x^3)/lambda)

Find the MLE of Lambda for a random sample of size n.


I began by taking the product of the function of lambda for all values of x

L(lambda, x1...xn) = (the product sign of i=1 to n) * f(xi, lambda) = (3/lambda) * x^2 * e^((-x^3)/lambda)

=> I take the derivative with respect to lambda...

dln (3/lambda^n * sum of i to n for xi^2 * e^((sum of i to n of -x^3)/(lambda^n)))

This is where I've either gone very wrong or I've just confused myself to no end.

I think you have to take the natural log with respect to lambda and solve for lambda, but the more I look at this problem and how I'm going about it the more I confuse myself.

If anyone could tell me how I could type this cleaner and look more like an actual problem on paper (with correct symbols, etc) I would be more than happy to go further, but this is basically where I'm stuck.

Any help is greatly appreciated.


I forgot to add the parameters

x > 0 and lambda > 0
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