Topics: Probability - Distribution Function


Let be a discrete random variable that handles the repetition of a Bernoulli trial until success is achieved. Let be the probability of success.

More specifically, let handle the number of trials needed to get one success (i.e. ).

For such an , we say that it has a shifted geometric distribution and write .

Density Function


Such an has the following density function:

Expected Value


When it comes to the expected value of such an , it is:



As for the variance of such an :

Moment-Generating Function


Finally, such an has the following moment-generating function: