Bernoulli vs. Poisson

Attribute	Bernoulli	Poisson
Number of trials	1	Infinite
Outcome	Success or failure	Count of events
Probability distribution	Discrete	Discrete
Mean	p	λ
Variance	p(1-p)	λ

Introduction

Bernoulli and Poisson distributions are two commonly used probability distributions in statistics. While they are both used to model random events, they have distinct characteristics that make them suitable for different types of data. In this article, we will compare the attributes of Bernoulli and Poisson distributions to understand their differences and similarities.

Definition

The Bernoulli distribution is a discrete probability distribution that represents the outcome of a single trial where there are only two possible outcomes - success or failure. The distribution is characterized by a single parameter, p, which represents the probability of success. On the other hand, the Poisson distribution is also a discrete probability distribution, but it is used to model the number of events that occur in a fixed interval of time or space. The Poisson distribution is characterized by a single parameter, λ, which represents the average rate of occurrence of the events.

Probability Mass Function

The probability mass function (PMF) of the Bernoulli distribution is given by P(X=x) = p^x * (1-p)^(1-x), where x is the outcome of the trial (0 for failure, 1 for success). The PMF of the Poisson distribution is given by P(X=x) = (e^(-λ) * λ^x) / x!, where x is the number of events that occur in the interval. The PMF of the Bernoulli distribution is a simple formula that depends on the probability of success, while the PMF of the Poisson distribution involves the exponential function and the factorial.

Mean and Variance

The mean of the Bernoulli distribution is E(X) = p, which represents the probability of success. The variance of the Bernoulli distribution is Var(X) = p(1-p), which is a measure of the spread of the distribution. In contrast, the mean of the Poisson distribution is E(X) = λ, which represents the average rate of occurrence of the events. The variance of the Poisson distribution is Var(X) = λ, which is equal to the mean. This property of the Poisson distribution is known as equidispersion.

Applications

The Bernoulli distribution is commonly used to model binary outcomes, such as success or failure, heads or tails, etc. It is often used in experiments where there are only two possible outcomes. For example, it can be used to model the outcome of a coin toss or the success of a marketing campaign. On the other hand, the Poisson distribution is used to model the number of events that occur in a fixed interval of time or space. It is often used in scenarios where the events occur independently at a constant rate, such as the number of phone calls received by a call center in an hour.

Relationship

While the Bernoulli and Poisson distributions are used for different types of data, there is a relationship between them. In fact, the Poisson distribution can be derived from the Bernoulli distribution under certain conditions. Specifically, if we have a large number of Bernoulli trials with a small probability of success, then the number of successes in a fixed interval of time or space can be approximated by a Poisson distribution with parameter λ = np, where n is the number of trials and p is the probability of success.

Conclusion

In conclusion, the Bernoulli and Poisson distributions are two important probability distributions that are used in statistics to model random events. While the Bernoulli distribution is used for binary outcomes with a single trial, the Poisson distribution is used for the number of events that occur in a fixed interval. Understanding the differences and similarities between these distributions is crucial for choosing the appropriate model for a given dataset.