# TI-BASIC:Poissoncdf

**Command Summary**

Calculates the Poisson cumulative probability for a single value

**Command Syntax**

poissoncdf(*mean*, *value*)

**Menu Location**

Press:

- 2ND DISTR to access the distribution menu
- ALPHA C to select poissoncdf(, or use arrows.

Press ALPHA D instead of ALPHA C on a TI-84+/SE with OS 2.30 or higher.

TI-83/84/+/SE

2 bytes

This command is used to calculate Poisson distribution cumulative probability. In plainer language, it solves a specific type of often-encountered probability problem, that occurs under the following conditions:

- A specific event happens at a known average rate (X occurrences per time interval)
- Each occurrence is independent of the time since the last occurrence
- We're interested in the probability that the event occurs at most a specific number of times in a given time interval.

The poissoncdf( command takes two arguments: The *mean* is the average number of times the event will happen during the time interval we're interested in. The *value* is the number of times we're interested in the event happening (so the output is the probability that the event happens at most *value* times in the interval). Note that you may need to convert the mean so that the time intervals in both cases match up. This is done by a simple proportion: if the event happens 10 times per minute, it happens 20 times per two minutes.

For example, consider point on a city street where an average of 5 cars pass by each minute. What is the probability that in a given minute, no more than 3 cars will drive by?

- The event is a car passing by, which happens at an average rate of 5 occurences per time interval (a minute)
- Each occurrence is independent of the time since the last occurrence (we'll assume this is true, though traffic might imply a correlation here)
- We're interested in the probability that the event occurs at most 3 times in the time interval.

The syntax in this case is:

:poissoncdf(5,3

This will give about .265 when you run it, so there's a .265 probability that in a given minute, no more than 3 cars will drive by.

# Formulas

The poissoncdf( command can be seen as a sum of poissonpdf( commands:

<math> \definecolor{darkgreen}{rgb}{0.90,0.91,0.859}\pagecolor{darkgreen} \operatorname{poissoncdf}(\lambda,k)=\sum_{i=0}^k \operatorname{poissonpdf}(\lambda,i) = \sum_{i=0}^k \frac {e^{-\lambda} \lambda^i}{i!} </math>

We can also write the poissoncdf( command in terms of the [wikipedia:Incomplete_gamma_function incomplete gamma function]:

<math> \definecolor{darkgreen}{rgb}{0.90,0.91,0.859}\pagecolor{darkgreen} \operatorname{poissoncdf}(\lambda,k)=\frac{\Gamma(k+1,\lambda)}{k!} </math>