# Estimating Conversions with Poisson Distributions

### More Random Events

Following up on last week’s post on normal distributions, this week we’re going to show off another common probability distribution.

Let’s say you typically sell 5 widgets per day. How likely is it that you’ll sell 5 widgets tomorrow? What about between 4 and 6 widgets tomorrow? Obviously we can’t just guess randomly. And the normal distribution won’t help either.

Fortunately, this is what the Poisson Distribution is for. Its formula is:

Our Poisson Distribution formula takes 3 inputs:

: Our known rate, in this case 5.**R**: Euler’s Number, 2.71828.**e**: tomorrow’s expected rate.**k**

### Breaking It Down In SQL

Let’s start by plotting the Distribution over several expected conversion rates (* k*) per day. As before, we’ll use

*for convenience, so MySQL and Redshift users should try out alternatives.*

**generate_series**withnumbersas(

select*fromgenerate_series(0,20) k

)

select

k,

(power(2.71828,-5)*power(5, k)/(k!)) pr

fromnumbers

orderby1

Using our favorite visualization tool, this gives us a graph that looks like this:

In other words, there’s an 18% chance that we’ll sell exactly 5 widgets tomorrow. The chances of selling more than 5 decrease rather quickly. There is virtually no chance of selling 15 widgets tomorrow.

### Asking Specific Questions

At the top of the post, we asked how likely it was that we’d sell between 4 and 6 widgets tomorrow. To ask a specific question like that, we need only add a * where* clause:

wherekbetween4and6

Putting it all together, we get SQL that looks like this:

withnumbersas(

select*fromgenerate_series(0,20) k

)

select

sum((power(2.71828,-5)*power(5, k)/(k!)) pr

fromnumbers

wherekbetween4and6

orderby1

In this case, there’s a 50% chance that we’ll sell between 4 and 6 widgets. Not bad!