This is used when target variable is 0, 1, 2, … and represents a count of some number of events occuring in a given unit of time or space. Here’s the Poisson distribution first:
Poisson Distribution
The Poisson distribution models the number of events occurring in a fixed interval of time, space, or opportunity, when events happen independently at a constant average rate.
Formula: For a nonnegative integer \(y = 0, 1, 2, \dots\), the probability is \[\Pr(Y = y) = \frac{\mu^y e^{-\mu}}{y!},\] where \(\mu > 0\) is both the mean and the variance.
- Key properties:
- Mean: \(\mathbb{E}[Y] = \mu\)
- Variance: \(\operatorname{Var}(Y) = \mu\)
- The distribution is skewed for small \(\mu\), and becomes more symmetric as \(\mu\) grows.
- Interpretation:
- \(Y\) counts how many times an event happens.
- \(\mu\) is the expected number of events in the interval.
- Example: number of emails received in an hour, number of accidents at an intersection in a day, number of awards a student earns in a year.