Not really a rule....
But the multiplying of distributions to get MLE's is a bit difficult, so instead must people take the log of the ditribution as this is a valid approach.
So say you are looking at Poisson pdf:
pdf = exp(-mu)*mu^X / X!
Then look at the MLE for say n observations - if we multiply this dist n times, we can write this expression using the pie symbol which means multiplication in this case:
π exp(-mu)*mu^Xi / Xi! ...., i = 1,2,3.......,n
Now when we take the log of something, we do it for each part of the eqn on either side of a multiplication or division operator. We then swap any multiplication signs for +'s, and any divisors for -'s.
E.g.
log(pdf) = log( exp(-mu)*mu^X / X!)
= log(exp(-mu)) + log(mu^X) - log(X!)
= -mu + Xlog(mu) - log(X!)
Now if we have put the pie operator inside our log expression, when we log the eqn, any xi's will be summed:
Log(∏ exp(-mu)*mu^X / X!))
= -n*mu + ∑Xilog(mu) - log(∑Xi!)
Then to find the MLE estimator we differentiate w.r.t. the parameter we are interested in, in this case mu :
d(-n*mu + ∑Xilog(mu) - log(∑Xi!)) / dmu = -n + ∑Xi / mu
Then set this eqn equal to zero and solve for mu to get the MLE:
-n + ∑Xi / mu = 0
mu = ∑Xi / n
i.e. MLE(mu) = ∑Xi / n
