Poisson

Represents the number of events occurring within a fixed interval of time or space. It is used for modeling the number of times an event happens in a given period of time, distance, area, or volume when these events occur with a known constant mean rate and independently of the time since the last event.

Create a new Poisson distribution

Poisson X = Poisson(lambda); 

creates a discrete Poisson probability distribution.

Distribution properties:

Property	Type/Return type	Description
lambda (λ)	double	Average number of events in the interval (also known as the rate parameter)
E()	double	Returns the expected value of average number of events in the interval.
D()	double	Represents variance, that measures the spread of dispersion of the random variable around its expected value E(X).
P(int k)	double	Probability of observing exactly 𝑘 events in an interval: P(X = k)
P_LT(int k)	double	Probability of observing less than 𝑘 events in an interval: P(X < k)
P_LTE(int k)	double	Probability of observing less than or exactly 𝑘 events in an interval: P(X ≤ k)
P_HT(int k)	double	Probability of observing more than 𝑘 events in an interval: P(X > k) = P(X ≤ k)
P_HTE(int k)	double	Probability of observing more than or exactly 𝑘 events in an interval: P(X ≥ k) = P(X < k)

Example

Example of a poisson distribution

//Example: A bookstore expects 3 visitors per hour. What's the probability 
//of encountering 5 visitors in an hour?
double lambda = 3; //3 visitors per hour
Poisson X = Poisson(lambda);
std::cout << X.P(5) << std::endl; //0.100819

Common errors

Poisson X = Poisson(5); 
std::cout << X.P(-1) << std::endl; //Error: Random variable cannot be less than 0.  

Good to know: k cannot be negative.