In probability theory and statistics, λ (lambda) is a parameter that represents the average rate or average number of events occurring in a fixed interval in a Poisson distribution.

In the context of the Poisson distribution, λ determines the shape and characteristics of the distribution. It represents the average number of events that occur per unit of time, space, or any other fixed interval.

The value of λ can be any positive real number, indicating the intensity or frequency of the events. For example, if λ = 2, it means that on average, 2 events occur in the specified interval.

The value of λ influences various properties of the Poisson distribution, such as the mean, variance, and probability mass function.

The probability mass function of a Poisson distribution is given by:

\( P(X = k) = (\frac{e^{-\lambda} \cdot \lambda^k}{k!}) \)

Here, λ appears as the exponent in the exponential term and as the parameter for the mean and variance. The value of λ determines the shape and characteristics of the Poisson distribution, such as the average number of events occurring in a given time period.