Question

The geometric distribution with parameter p represents the number of failures in asequence of independent Bernoulli trials before a success occurs. Show that this distribution is of the exponential family

Answer 1

The geometric distribution has the probability density function:

f(x | p) = (1-p)^x-1*p = exp[x*log(1-p) + log(p/1-p)]

And this expression satisfies the definition of exponential family:

f(x | p) = h(x) * exp[a(p)*T(x) + b(p)]

Note that in our case h(x) = 1