POISSON

Calculates values for a Poisson distribution.

Syntax:

POISSON(x, λ, mode)

The Poisson distribution is a discrete probability distribution giving the probability that x events occur in a certain time, where events occur independently, and where on average λ events are expected. x should be >=0 and λ should be >0. x should be integer.

If mode is 0, POISSON calculates the probability density function of the Poisson distribution:

$\frac{e ^{- λ} λ ^{x}}{x !}$

If mode is 1, POISSON calculates the cumulative distribution function of the Poisson distribution:

$\sum_{k = 0}^{x} \frac{e ^{- λ} λ ^{k}}{k !}$

Example:

POISSON(8; 9; 0)

returns approximately 0.1317, the probability that exactly 8 events occur in a time period where you expect 9 events.

POISSON(8; 9; 1)

returns approximately 0.4556, the probability that up to and including 8 events occur in a time period where you expect 9 events.

Application:

An application of the Poisson function can be found in a call center. Let's say a call center receives an average of 10 customer service calls per hour. We want to use the POISSON function to determine the probability of receiving a specific number of calls in a given hour.

The formula for the Poisson function is:

$P (X = x) = \frac{λ ^{x} e ^{- λ}}{x !}$

Where:

x is the number of events that occur in a specific period
λ is the average number of events per period
e is Euler's number (e≈2.71828)
x! is the factorial of x

Here's how we can use the POISSON function in our example:

Problem: A call center receives an average of 10 calls per hour. We want to find the probability of receiving exactly 7 calls in the next hour.

Parameters:

x=7 (the number of calls we want to find the probability for)
λ=10 (the average number of calls per hour)
mode = 0 (we want the probability density function, i.e., the probability of exactly 7 calls)

Using the POISSON function:

The POISSON function in a spreadsheet program would be entered as: POISSON(7, 10, 0)

Result: The function would return approximately 0.090079.

This means there is approximately a 9% chance of receiving exactly 7 calls in the next hour.

We can also use the POISSON function to calculate the cumulative probability.

Problem: What is the probability of receiving 7 or fewer calls in the next hour?

Parameters:

x=7
λ=10
mode = 1 (we want the cumulative distribution function, i.e., the probability of 7 or fewer calls)

Using the POISSON function:

The POISSON function would be entered as: POISSON(7, 10, 1)

Result: The function would return approximately 0.22022.

This means there is approximately a 22% chance of receiving 7 or fewer calls in the next hour.

Here is a table demonstrating the probabilities for a range of call counts, using the POISSON function:

	Number of Calls (x)	POISSON(x, 10, 0) - Probability of Exactly x Calls	POISSON(x, 10, 1) - Probability of x or Fewer Calls
	A	B	C
1	1	0.000453999	0.000499399
2	2	0.002269996	0.002769396
3	3	0.007566655	0.010336051
4	4	0.018916637	0.029252688
5	5	0.037833275	0.067085963
6	6	0.063055458	0.130141421
7	7	0.090079226	0.220220647
8	8	0.112599032	0.332819679
9	9	0.125110036	0.457929714
10	10	0.125110036	0.58303975
11	11	0.113736396	0.696776146
12	12	0.09478033	0.791556476
13	13	0.072907946	0.864464423
14	14	0.052077104	0.916541527
15	15	0.03471807	0.951259597
16	16	0.021698794	0.97295839
17	17	0.012763996	0.985722386
18	18	0.007091109	0.992813495
19	19	0.003732163	0.996545658
20	20	0.001866081	0.998411739

This table shows the probability of getting exactly a certain number of calls and the cumulative probability of getting that number of calls or fewer.