# MLE for mean in geometric distribution

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

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.