## 1.5. Standard Discrete Distributions

We now present some discrete distributions that frequently arise when studying networking problems.

### 1.5.1. Bernoulli Distribution

A discrete random variable *X* is called a **Bernoulli** random variable if it can take only two values, 0 or 1, and its probability mass function is defined as *p*(0) = 1 – *a* and *p*(1) = *a*. We can think of *X* as representing the result of some experiment, with *X*=1 being success, with probability *a*. The expected value of a Bernoulli random variable is *a* and variance is *p*(1 – *a*).

### 1.5.2. Binomial Distribution

Consider a series of *n* Bernoulli experiments where the result of each experiment is *independent* of the others. We would naturally like to know the number of successes in these *n* trials. This can be represented by a discrete random variable *X* with parameters (*n,a*) and is called a **binomial** random variable. The probability mass function of a binomial random variable with parameters (*n,a*) is given by

If we set *b* = 1 – *a*, then these are just the terms of the expansion (*a* + *b*)* ^{n}*. The expected value of a variable that is binomially distributed with parameters (

*n,a*) is

*na*.

### 1.5.3. Geometric Distribution

Consider a sequence of independent Bernoulli experiments, each of which succeeds with probability *a*. In section 1.5.2, we wanted to count the number of successes; now, we want to compute the probability mass function of a random variable *X* that represents the number of trials before the first success. Such a variable is called a **geometric** random variable and has a probability mass function

The expected value of a geometrically distributed variable with parameter *a* is 1/*a*.

### 1.5.4. Poisson Distribution

The **Poisson** distribution is widely encountered in networking situations, usually to model the arrival of packets or new end-to-end connections to a switch or a router. A discrete random variable *X* with the domain {0, 1, 2, 3,...} is said to be a Poisson random variable with parameter λ if, for some λ > 0:

Poisson variables are often used to model the number of events that happen in a fixed time interval. If the events are reasonably rare, the probability that multiple events occur in a fixed time interval drops off rapidly, due to the *i*! term in the denominator. The first use of Poisson variables, indeed, was to investigate the number of soldier deaths due to being kicked by a horse in Napoleon’s army!

The Poisson distribution, which has only a single parameter λ, can be used to model a binomial distribution with two parameters (*n* and *a*) when *n* is “large” and *a* is “small.” In this case, the Poisson variable’s parameter λ corresponds to the product of the two binomial parameters (i.e., λ = *n*_{Binomial}* *a*_{Binomial}). Recall that a binomial distribution arises naturally when we conduct independent trials. The Poisson distribution, therefore, arises when the number of such independent trials is large, and the probability of success of each trial is small. The expected value of a Poisson distributed random variable with parameter λ is also λ.

Consider an endpoint sending a packet on a link. We can model the transmission of a packet by the endpoint in a given time interval as a trial as follows: If the source sends a packet in a particular interval, we will call the trial a success; if the source does not send a packet, we will call the trial a failure. When the load generated by each source is light, the probability of success of a trial defined in this manner, which is just the packet transmission probability, is small. Therefore, as the number of endpoints grows, and if we can assume the endpoints to be independent, the sum of their loads will be well modeled by a Poisson random variable. This is heartening because systems subjected to a Poisson load are mathematically tractable, as we will see in our discussion of queueing theory. Unfortunately, over the last two decades, numerous measurements have shown that actual traffic can be far from Poisson. Therefore, this modeling assumption should be used with care and only as a rough approximation to reality.

It turns out that a Poisson random variable is a good approximation to a binomial random variable even if the trials are weakly dependent. Indeed, we do not even require the trials to have equal probabilities, as long as the probability of success of each individual trial is “small.” This is another reason why the Poisson random variable is frequently used to model the behavior of aggregates.