Sum of two Poisson distributions

Introduction to Sum of two Poisson Distributions

• The Poisson distribution is a probability distribution that represents the number of events in a fixed interval of time or space.

• It is commonly used in statistics to model random variables which depict the number of times an event happened in a time or space window, such as number of emails received in an hour or number of vehicles passing through a checkpoint in a day.

Understanding the Sum of two Poisson Distributions

• If two independent random variables X and Y follow Poisson distributions with parameters λ and μ respectively, then the sum Z (where Z = X + Y) will also follow a Poisson distribution.

• The parameter for the sum Z is the sum of the parameters of X and Y. Therefore, if X ~ Poisson(λ) and Y ~ Poisson(μ), then Z = X + Y ~ Poisson(λ + μ).

Properties of the Sum of Poisson Distributions

• The expectation (mean value) of the sum of two Poisson distributions is the sum of the expectations of the individual distributions. If X ~ Poisson(λ) and Y ~ Poisson(μ), then E(Z) = E(X) + E(Y) = λ + μ.

• The variance of the sum of two independent Poisson distributions is the sum of their variances. If X ~ Poisson(λ) and Y ~ Poisson(μ), then Var(Z) = Var(X) + Var(Y) = λ + μ.

Applications of the Sum of Poisson Distributions

• The sum of two Poisson distributions can provide insight on the behaviours of combined independent events. For example, if X is the number of customer calls received at a call centre in the morning and Y is the number in the afternoon, then Z represents the total number of calls for the entire day.

• Summing Poisson distributions can also help in determining the total number or frequency of events happening over larger time frames or bigger spaces by simply summing the distributions over the required intervals.

Further Considerations

• Understanding the properties and behaviour of summed Poisson distributions is integral for statistical modelling and prediction when dealing with independent events occurring over time or space.

• Proficiency in calculating the sum of Poisson distributions will aid in tackling problems where several independent Poisson processes contribute to a single overall output.