# What is the name of this formula derived from the Poisson distribution?

I am learning about the Poisson distribution in this document and its link reference.

This is the key formula to compute the Poisson distribution:

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

I saw another related formula somewhere.

$$\sum\limits_{k = x}^{+ \infty} \frac{\lambda^k e^{-\lambda}}{k!}$$

Is there a name for this formula?

## 4 Answers

The first formula is the probability mass function (PMF) of the Poisson distribution and the second one is the survival function of this distribution. The first one gives you the probability that the Poisson random variable (which can take integer values) will equal $$k$$ and the second one the probability that it will be greater than or equal to $$k$$.

The Taylor series for the function $$g(\lambda) = e^\lambda$$ is $$e^\lambda = \sum_{k=0}^\infty \frac{\lambda^k}{k!}.$$ By rearranging, you can verify that the sum of the probabilities of all outcomes for a Poisson($$\lambda$$) random variable $$X$$ is $$1$$. $$\sum_{k=0}^\infty P(X = k) = \sum_{k=0}^\infty e^{-\lambda} \frac{\lambda^k}{k!} = 1.$$

To add on the answers by angryavian and Rohit Pandey, the complementary of the second formula is also the cumulative distribution function (CDF): $$F(x-1) = P(X \leq x-1) = \sum_{k = 0}^{x-1} \frac{\lambda^k e^{-\lambda}}{k!} = 1 - \sum_{k = x}^{+ \infty} \frac{\lambda^k e^{-\lambda}}{k!} \ .$$

To add on the answers by angryavian, Rohit Pandey and Ertxiem, the second formula is the complement of the CDF of Poisson PMF for "x-1"

$$1 - F(x-1) = \sum_{k = x}^{+ \infty} \frac{\lambda^k e^{-\lambda}}{k!}$$

which means the probability of at least $$x$$ observations