Topics: Probability


Let be a discrete random variable that handles the number of events that occur a given time or space interval, knowing that these events occur with a known constant mean rate and independently of the time since the last event.

For such an , we write:

…where is the parameter that represents the number of times that an event is expected to occur in a given interval (). We use this interval and its as our main references when talking about a Poisson distribution.

Density Function


With an such that , its density function is:

Expected Value


Such an has a very simple expected value:



Such an also has a very simple variance:

Notice it’s the same as its expected value.

Moment-Generating Function


As for the moment-generating function of such an , it is:


Let be the number of messages that are rejected by a web server in a second. We know that 5 messages are expected to be rejected every second (i.e. we know that ).

We’ll calculate:

  1. The probability that exactly 3 messages are rejected in a second
  2. The probability that at most 2 messages are rejected in a second.

Exactly 3 messages

Since is discrete, we can simply evaluate the density function at :

At most 2 messages

From the idea of a distribution function, we have that: