I've tried for a long time to think of a way to solve this.
This problem is part of an assignment, we were given sample data and some model pdf functions to check which fits best for the data. We're supposed to use the point estimates of μ and σ to estimate the parameters a and b.
The below pdf is the one I'm having trouble with:
Question: Now how would I go about estimating a and b here? The way they phrase it sounds like we should use
To calculate a and b? But that doesn't work does it? Or am I just failing to solve it properly that way? Is the only proper way to use MLE? But that seems hard to use in this case. The course isn't very advanced so I wouldn't expect the solution to be highly advanced.
Note: the constants a and b are both positive according to instructions
The first thing you need to do is find a proper normalization for the pdf, call
$$ f(x) = c x(x + b)e^{-a x} $$
where $a$ is a constant such that
$$ \int_0^{+\infty}dx\;f(x) = 1 = c\frac{2+ab}{a^3} \quad\Rightarrow\quad c = \frac{a^3}{2 + ab} $$
Now, you can find the expected value of $x$,
$$ \mu = \int_0^{+\infty}dx\;xf(x)= \frac{6 + 2 a b}{2 a + a^2 b} \tag{1} $$
and variance
$$ \sigma^2 = \int_0^{+\infty}dx\;(x - \mu)^2f(x) = 2 \frac{6 + a b (6 + a b)}{a^2 (2 + a b)^2} \tag{2} $$
if you manage to find a point estimation of $\mu$ and $\sigma$ you can use Eqs. (1) and (2) to find $a$ and $b$
If you want to use maximum likelihood instead, imagine you have now a set of points of the form $\{x_i\}_{i=1}^n$, define the likelihood as,
$$ \mathcal{L}(a, b) = \prod_{i=1}^n f(x_i;a,b) $$
or taking log:
$$ \ln\mathcal{L}(a,b) = n\ln\frac{a^3}{2 + ab} + \sum_{i=1}^n \ln x_i + \sum_{i=1}^n \ln(x_i + b) - \sum_{i=1}^n ax_i $$
You then need to optimize $\ln \mathcal{L}$ agains $a$ and $b$