Tech Tips: Poisson Distributions

Calculating $P(X=x)$ when $X$ follows a Poisson Distribution

Suppose one wishes to find the Poisson probability of seeing exactly $k$ occurrences of some event within some well-defined interval, where the mean number of occurrences in that interval is expected to be $\lambda$. That is to say, we seek

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

To do this one should ...


Calculating Cumulative Probabilities when $X$ follows a Poisson Distribution

Suppose one wishes to fine the cumulative Poisson probability of seeing $k$ or fewer occurences of some event within some well-defined interval or range, where the mean number of occurrences in that interval is expected to be $\lambda$. That is to say, we seek $$P(X \le k) = P(0) + P(1) + P(2) + \cdots + P(k) = \sum_{0 \le i \le k} \frac{e^{-\lambda} \lambda^k}{k!}$$ To do this, one should ...


Simulating Random Variables following Poisson Distributions

To generate $n$ realizations of a random variable that follows a Poisson distribution, counting the number of occurences of some event within some well-defined interval or range, where the mean number of occurences in that interval is expected to be $\lambda$. ...