Suppose that a random experiment has two possible outcomes, success with probability $p$ and failure with probability $q=1-p$. The experiment is repeated until a success happens. The number of trials before the success is a random variable $X$ with density function

The distribution function determined by $f(x)$ is called a geometric distribution with parameter $p$ and it is given by

The picture shows the graph for $f(x)$ with $p=1/4$. Notice the quick decreasing. An interpretation is that a long run of failures is very unlikely.

We can use the moment generating function method in order to get the mean and variance. This function is

The last expression can be simplified as

The first derivative is

so the mean is

In order to find the variance, we use the second derivative and thus

and therefore the variance is