Discrete probability distribution

A random variable can be discrete if it has:

• a finite number of values; or
• an infinite but countable number of values.

A discrete probability distribution, therefore, is the probabity distribution of a discrete random variable (one that has a finite number of values). Two common distributions include the:

• binomial distribution; and
• Poisson distribution.

Binomial distribution

A binomial distribution is only valid if four conditions are met (2pin):

• Each trial of the experiment must result in only two (2) outcomes (e.g., yes or no; success or failure).
• The probability (p) of a successful outcome is constant for each trial.
• The trials are independent (i) of one another — i.e., the result of one trial does not affect another.
• There are a finite number (n) of trials in the experiment.

In an experiment of $n$ independent trials,

• $p$ is the probability of a successful outcome; and
• $X$ is the random variable.

The notation to describe a binomial distribution is:

$$X\sim B(n,p)$$

Provided a binomial distribution, the probability when $X=a$ is denoted by the following formula:

$$P(X=a)={}^n{C}_a{p}^a{q}^{n-a}$$

(where $q=1-p$)

The expected value or mean of the distribution is denoted by the following:

$$E(X)=np$$

The variance can be calculated by the following:

$$Var(X)=npq$$

Poisson distribution

A Poisson distribution is only valid if four conditions are met:

• The event must occur randomly.
• The probability an event will occur in a certain time interval is proportional to the size of the interval.
• The number of events occurring in a unit of time is independent of the number of events that occur in other units of time.
• In a very small interval, the probability that two or more events will occur tends to zero.

A distribution that describes the number of times an event will occur randomly in:

• a given interval of time; or
• a given space (e.g., area, volume, weight, distance).

The notation to describe a Poisson distribution is:

$$X\sim P_O(\lambda )$$

(where $\lambda$ is any number more than zero, often the average for the given time)

Provided a Poisson distribution, the probability when $X=a$ is denoted by the following formula:

$$P(X=a)=e^{-\lambda }\frac{{\lambda }^a}{a!}$$

The expected value or mean of the distribution is $\lambda$. The variance of the distribution is $\lambda$.

Poisson approximation from binomial distribution

When a binomial distribution has a high number of independent trails ($n$) and a low probability ($p$), we can approximate the binomial distribution into a Poisson distribution.

Expressed mathematically, when $n\to \infty ,p\to 0$,

$$X\sim B(n,p)\approx X\sim P_O(np)$$