# What do we know about $Z$ if $Z = U - V$ where $U$ and $V$ are indep. and both exponentially distributed

Assume now that Ali and Berta independently assign exponentially distributed scores, with rates $$a$$ and $$b$$, respectively. But they only record the difference between the two scores (Ali’s score minus Berta’s score, say.) What is the maximum likelihood estimator for $$a$$ and $$b$$, using only the score differences?

In a previous part I calculated the MLE of the RV $$X$$ that was the sum of their scores that was exponentially dist with parameter $$\lambda$$ as:

$$\hat{\lambda} = \frac{n}{\sum_{i=1}^{n} x_i}$$

Note that in that part we only observed $$X$$ and nothing about their scores or score rates individually.

Can I just consider this same result for my new RVs $$U$$ and $$V$$ with their parameter's $$a$$ and $$b$$? How would you suggest approaching this new RV $$Z$$ that is their difference?

I see here Find the distribution of $Z=X+Y$ where both $X$ and $Y$ are exponentially distributed. that there might be something to the new variable having the gamma dist. Is this a correct path to take?

I'm just looking for a helpful push or maybe a resource that shows me the right direction.

Edit: $$U$$, $$V$$ $$\geq 0$$ sorry forgot to add this before.

Based on my calculation, the density of $$Z$$ would be:
$$f(v) = \frac{ ab}{a+b} e^{ -a|v|}$$ if $$v \ge 0$$
and $$f(v)= \frac{ ab}{a+b} e^{ -b|v|}$$ if $$v \le 0$$
• I forgot to add to the post that u and v are both $\geq 0$. But thank you for your time giving me some answer. Can I ask how you got started making that density calculation? Knowing what we know about $U$ and $V$ what do you then know about $Z$ that makes you able to calculate density? Nov 7 '20 at 1:37
• Nothing in particular. I just calculate it. So, I just call out any bounded continuous function $f$ then I use the change of variable to simplify the formula of $\mathbb{E}( f(Z))$. Nov 7 '20 at 1:41