I was working on the task where it was asked to found the pdf of the sum of 2 rv having uniform distribution, using transformation theorem (representing bandwidths of routers, and the bandwidth of the system consisting of routers):
$f_{S}(s) = \frac{1}{\theta}, for 0 < s < \theta$
So, I got the pdf of $S_1 + S_2$ and got the following:
$f_{T}(t) =
\begin{cases}
\dfrac{t}{2\theta^2}, & 0 < t < \theta \\
-\dfrac{t-2\theta}{2\theta^2}, & \theta < t < 2\theta \\
0, & \text{otherwise}
\end{cases}$
Now, I am asked to estimate $\theta$ using MLE method, for the experiment results 59, 86, 71, 36, 28.
I am confused how to work with different ranges?
The pdf could be converted to
$f_{T}(t) = \frac{t}{2\theta^2}\mathbf I_{(0,\theta)} - \frac{t-2\theta}{2\theta^2} \mathbf I_{(\theta,2\theta)}$
where $\mathbf I$ is indicator function. How I should handle Indicator functions in MLE? Assuming the experiments are independent, I should multiply pdfs for each experiments to get the MLE, but how to be with Indicator functions? should I consider each cases separately, like:
- θ < 59,
- θ < 71,
- θ < 86,
- θ > 86,
And for each case to calculate MLE and take the maximum one?