Bernoulli, Binomial, Poisson & Exponential Distributions Explained with Examples

1 Bernoulli Distribution

1.1 Definition

A Bernoulli trial is a random experiment with exactly two possible outcomes, success or failure, where the probability of success is constant for each trial. The Bernoulli distribution is the discrete distribution of a Bernoulli trial.

Here X denotes the random variable, x denotes a possible value of the random variable, and p is the probability of success in the trial.

1.2 Example

Viewing a single lottery ticket purchase as either winning or not winning can be modeled with a Bernoulli distribution.

2 Binomial Distribution

2.1 Definition

In n independent repeated Bernoulli trials, let the probability of event A occurring in each trial be p . Let X denote the number of times event A occurs in the n trials. Then X can take values 0, 1, ..., n . For each k , the event {X = k} means “event A occurs exactly k times in n trials”, and the discrete probability distribution of X is the Binomial Distribution.

The Binomial distribution has four characteristics:

n identical trials

Each trial results in either “success” or “failure”

The probability of “success” remains constant

Trials are mutually independent

The probability mass function (PMF) of the Binomial distribution is:

2.2 Example

Consider a simple example to illustrate the Binomial PMF.

Each month a private banker meets 50 people to inquire about loans. Based on experience, 30% of them have a bad credit record. Compute the probabilities that the number of people with bad credit records is 1, 2, ..., 50.

The probability that the banker encounters exactly 14 customers with bad credit records is calculated using the Binomial formula.

The figure below shows the probability distributions for different values of p (0.1, 0.3, 0.5, 0.7, 0.9).

3 Poisson Distribution

The Poisson distribution describes the probability of a given number of events occurring in a fixed interval of time or space, based on the average rate of occurrence. It is often used for low‑demand inventory control, such as estimating how many lobsters a restaurant should stock to meet demand with a 95% confidence level.

When n is large and p is small, the Binomial distribution converges to the Poisson distribution.

3.1 Example

For a typical trader, the probability of encountering a “black‑swan” event in a single trade is 1/1000. If a trader makes 20 trades per month, what is the probability of experiencing exactly 2 black‑swan events over 5 years?

This problem can be solved using the Binomial formula, but it can also be approximated by a Poisson distribution with parameter λ = 1.2 , meaning on average 1.2 black‑swan events occur every 5 years.

The figure below compares a simulated Binomial distribution (parameters matching the example) with a Poisson distribution of λ = 1.2 (ignore the y‑axis scale).

4 Negative Exponential Distribution

When event occurrences follow a Poisson process, the inter‑arrival times follow a Negative Exponential distribution. Its probability density function is:

f(t) = λ e^{-λ t}, t ≥ 0 , where λ is the same parameter as in the Poisson distribution, representing the average number of events per unit time.

5 Summary

This article introduced several interrelated probability distributions: Bernoulli, Binomial, Poisson, and Negative Exponential.

References

https://baijiahao.baidu.com/s?id=1703456602669206942&wfr=spider&for=pc

https://zhuanlan.zhihu.com/p/77003855