Poisson Distribution EXPLAINED in UNDER 15 MINUTES!

[Music]

all right folks it's the pon

distribution for today named after a

French dude that did a whole bunch of

stuff in stats but for our purposes we

only care about his work as it relates

to probability distributions so here's a

quick rundown firstly we know it's a

discreet dist distribution meaning

there's only a discrete set of values

that this distribution can take but what

does this distribution actually describe

well it describes the number of events

occurring in a fixed time interval or

region of opportunity so the classic

example is you know how many customers

does a bank teller get every hour so in

that instance there's a fixed time

interval of 1 hour and this distribution

might describe the number of events that

occurs with within that hour but I'm not

too sure that that many people are going

to bank tellers these days perhaps a

better modern day equivalent might be

the people arriving at an Apple Genius

Bar because their freaking iPhone 7

headphones don't

work back on topic the next feature

about the distribution is that it

requires only one parameter which is the

expected number of events per time

interval

Lambda so in this case I've put a

distribution here with Lambda equaling 3

so maybe that's three customers every

hour or whatever but the other thing to

note is that it's bounded by zero and

infinity so unlike the binomial

distribution if you've been watching

this series the pon actually continues

on Forever My Graph here stops at 10 but

there's theoretically a very small

probability of there being 10 events in

this time interval and there's also a

smaller probability of there being 11

and 12 and 13 Etc it's just I haven't

included them on the graph because they

become

negligible nonetheless it's a

theoretical difference between this and

the binomial distribution which needs to

be

appreciated now what are the assumptions

underlying the poson distribution

firstly the rate at which events occur

must be constant another way of saying

this is that the probability of an event

occurring in a certain time interval

should be exactly the same same for

every other time interval of that same

length the other assumption is that the

occurrence of one event does not affect

the occurrence of a subsequent event I.E

the events are independent so it

shouldn't matter if an event just

happened that shouldn't influence the

time interval till the next

event now these assumptions won't

necessarily hold in reality so it's

often good to appreciate when you're

using the pon distribution just how

relevant it is to the question at hand

and we'll see with some examples here

whether these assumptions are going to

break

down so the next thing we can learn

about the pon distribution is the pmf or

the probability Mass function now all

that means is the height of each of

these discrete outcomes or the

probability of getting each of these

discrete outcomes when the mean is three

in this case so for example if I wanted

to find the probability of getting say

five events happening in this time

interval I can use the formula subbing

in the value five for x and subbing in

the value three for Lambda because don't

forget three is still our mean or

expected number of events per this time

period so I can actually do this by hand

and in that case I'd get

0.101 but of course it is possible to

use the wonders of excel to do exactly

the same thing so if you use the pon

dois function now these are the new

statistical functions that were

introduced to excel in I think the 2013

version but they're all standardized now

which makes things really easy you might

find other sources giving you the old

formula here and that'll work too but I

think it's good to start using these dot

disc functions because you'll see

they're all exactly the same once you

get the handle on them so pan. dis

requires three different arguments the

first of which is the value we're

seeking the number of events for which

we're seeking the probability so if we

want five in that case we'll put five as

our first argument the second argument

requires the mean for the Plus on

distribution in this case that's three

and the third argument requires you to

tell it whether you want the cumulative

distribution which is called the CDF or

whether you want the probability Mass

function the pmf and of course we want

the latter so to write false that tells

it that we don't want the cumulative

ative distribution we want the pmf so

it'll give us .11 as well so there's a

10% chance roughly 10.1% chance of

getting five events occurring in this

time interval all right so let's talk

about the CDF now the cumulative

distribution function now that's not the

height of a certain individual discrete

outcome that is the cumulative

distribution so all of the heights put

together up until that point and then

there is a formula for that involving a

gamma distribution and all this stuff

which look you're not going to really

need to know unless you're doing higher

order statistical stuff but for this

purpose we can use Pon doist again but

make sure we write true as that third

argument and in that case it's going to

sum up for us all of these bars up to

and including the outcome where there

are five

events so it'll sum up all of those five

and we get 0 point

916 the other unique thing about The

Poon distribution is that its expected

value which we kind of need to be told

is actually equal to its variance so

Lambda is also the variance of the

distribution so here we've got a mean of

three and we'd also have a standard

deviation of the square root of three

okay remembering that the standard

deviation is the square root of the

variance

and what I've done now is just provided

for you a couple of poson distributions

for differing values of Lambda just so

you get a sense of what they look like

so this is a Pon distribution with

Lambda that's the expected value being

one and this one is for where Lambda is

two here's Lambda being three and four

and five now of course in each of these

circumstances the distribution continues

Beyond 10 but I've just left the scale

constant so you can kind of get a sense

of how these flow from 1 2 3 4 and

5 but be aware too that the mean Lambda

doesn't need to be an integer value you

can also have a Plus on distribution

with a mean of 3.61 for example or even

a mean of 0.5 so there's no requirement

for that Lambda to be a full whole

number all right so it's time for you to

do a question let's give this a read

exclusive Vines import Argentinian wine

into Australia and they've begun

advertising on Facebook to direct

traffic to their website where customers

can order wine online the number of

click-through sales from the ad is Pon

distributed with a mean of 12

clickthrough sales per day Okay so we've

got a mean of 12 that will be our Lambda

value now I've got three questions for

you and I'm hoping that you can pause

the video here and give these a go and

then see if we get the same answer but

we're after the probability of getting

exactly 10 click-through sales in the

first day at least 10 click-through

sales in the first day and then more

than one sale in the first hour so see

how you go with those and I'm also going

to give you a bonus question do you

think the pon distribution is actually

appropriate for this scenario in reality

so hopefully you can think about those

assumptions and figure out whether they

would hold in this case so here's the

answer to part A the probability of 10

clickthrough sales in the first day is

equivalent to the height of this bar

here now this is a plus on distribution

where the mean is 12 Lambda is 12 of

course it goes a little Beyond in this

direction down to zero and on this

direction it goes up to Infinity but how

do we find the probability of that bar

we can use the formula for the pmf and

just Sub in those values for 12 being

Lambda and 10 being X and we we get a

value of

0.105 which would be the same result if

you used equals plus

on. and subbed in 10 12 and

false so there's a 10.5% chance of

getting 10 click-through sales in that

first day so there's the probability

Illustrated on the

plot all right Part B what's the

probability of at least 10 clickthrough

sales on the first

day so how do we find the probability of

X being greater than or equal to

10 now that's equivalent to this whole

yellow area over here if we sum up all

of those together going from 10 up until

Infinity well unfortunately in Excel

there's no way of finding the

probability of getting a value or higher

so we're going to have to use the CDF

which is the probability of getting a

value or lower and subtracting it from

one now here's the trick we're actually

going to go one minus plus on dist n is

the value of Interest we're going to use

here because if you think about it here

are those green

bars we're going to go one minus the

probability of all of these green bars

put together which is 9 and

below so you have to use your brain a

little bit with some of this stuff

knowing that it said at least 10

clickthrough sales we know we were after

10 and above which is 1 minus the

probability of N and

Below but putting in the appropriate

formula here we can get

0.758 and hopefully you got that exact

answer what about the probability that

we have more than one click-through sale

in the first

hour well this is where the properties

of a Pon distribution show themselves if

we know there's an average of 12

click-through sales in the first day if

it's truly A Plus on distribution the

mean number of sales per hour will be

0.5 because all we do is just divide by

the total number of hours so this

becomes our new value of

Lambda so this is the distribution now

where we've got a Lambda value of

0.5 so most of the distribution is going

to be down here at 0 and 1 because we're

only expecting 0.5 sales per hour so

we're most likely to get zero sales in a

given hour potentially we can get one

and then it becomes less likely to get

two three and very unlikely to get four

five and six and Beyond so really we're

after this shaded yellow region which

will include all of those values from

Two and

Beyond to get that it's going to be very

much like the last example we're going

to go 1 minus plus onist where we're

taking the cumulative distribution so

putting true in that third argument but

we're doing it from one where the mean

is 0.5 so we're going to be subtracting

from one these two bars

here which is

0.090 so there's only a 9% probability

of getting more than one click-through

sale in the first hour and I'll just

reiterate that it's it's very important

to read strictly what was written in the

question here because it says more than

one click through sale if it said one or

more we'd actually get a different

answer because we'd be after this

probability as well so this would also

be yellow where X is

one all right so how did you go

hopefully you got those same

answers question D the bonus question

asks you to think about what a Pon

distribution kind of is and whether it's

appropriate for this scenario in

reality now I'll return you to our

assumption where it said that the the

rate at which events occur must be

constant so in other words no interval

can be more likely to have an event than

any other interval of the same

size now if you're dealing with clicks

on Facebook over the course of a day

it's very unlikely for it to be a

constant

rate how many people do you think are

going to be clicking on ads for wine at

about 2:00 a.m. in the morning well

actually that's probably quite a few how

many people are going to be looking at

it at say 6: a.m. in the morning might

be a better

question not very many right so

irrespective of when in the day people

would be likely to click on this you can

you get the sense that people's usage of

Facebook would differ throughout the

hours of the day and also their

likelihood to be attracted by an ad for

wine so in reality it would not really

be a Pon distribution but nonetheless it

is often good to use these types of

distributions as it can still give a

decent picture of what's going on feel

free to click through to the next video

in the series where I deal with the

hyper geometric

distribution and any poker players might

be interested in that one that is the

classic poker hand

distribution but yep I'll just leave

these up here and if you want to keep in

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