Deriving Poisson Equation

Poisson distribution is defined as,

$P(x; \lambda) = \frac{e^{-\lambda} \lambda^x}{x!}$

where,

• $ P(x;λ) $ is the probability that an event occurs x times in the given interval,
• $ $ is the expected rate/probability of an event occuring.

In Binomial distribution, probability that one of the two events(p and q) occurs x times out of n trials is, $P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x}$

where q = 1 − p. Now, if λ is the expected number of successes then, $p=\frac{\lambda}{n}$.

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