MLE for mean in geometric distribution

Question

i have a random sample of n observations from geometric distribution pdf $p q^x-1$ but i m not able to find the

1) maximum likelihood estimator of the mean ,

2) MLE of P using exponential family

as i found MLE for P but confused about MLE for mean and for P of exponential family

Answer 1

Hint:

One form of the geometric distribution has probability mass function (not density) $pq^{x-1}$ on the non-negative integers

in which case the mean is $\mu=\frac{q}{p}$ so $p = \frac{1}{\mu+1}$ and $q=\frac{\mu}{\mu+1}$ and the probability mass function is $\frac{\mu^{x-1}}{(\mu+1)^x}$

so you can find the likelihood of your $n$ observations as a function of $\mu$, $n$ and $\sum x_i$, and you can find the value of $\mu$ which maximises this.

You should also be able to find a simple relationship between your two answers.