Discrete probability distribution

Last updated Oct 31, 2022

• Mathematics

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.

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: $$ P(X = a) = {^n}C_a \space p^a \space q^{n-a} $$ (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: $$ Var(X) = npq $$

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: $$ P(X = a) = e^{-\lambda} \space \frac {\lambda^a} {a!} $$ The expected value or mean of the distribution is $\lambda$. The variance of the distribution is $\lambda$.

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 $n \to \infty, \space p \to 0$, $$ X \sim B(n, p) \approx X \sim P_O(np) $$