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Like binomial and Bernoulli random variables.

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A Poisson random variable is another type of discrete random variable, because again with Poisson,

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we're focusing on a countable number of outcomes.

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So we'll always treat Poisson variables as discrete, not continuous.

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Now the difference between a Poisson random variable and a binomial random variable.

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If we remember about the binomial random variable, we indicate a probability of success and then we're

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looking for the probability of K successes in n trials.

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So a binomial random variable is always indicated this way B of NP or sometimes capital X of NP, where

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N is the number of trials for performing.

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That's always a fixed number of trials and P is the probability of success on any particular trial.

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And when we calculate probability, we're looking at the probability of a certain number of successes

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in those end trials.

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So if we're running an equals 20 trials and our probability of success is 30% or 0.3, then we could

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go on to calculate probability around seeing any number of successes from K equals zero successes all

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the way up to K equals 20 successes.

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Since we're running an equals 20 trials.

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But with the Poisson variable, we have this idea that the rate of occurrence of some event or object

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or something that's going on per unit of something else we call lambda, this Greek letter lambda,

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and then we're looking for or we're investigating probability of a certain number of occurrences or

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successes over one unit where these two units are referring to the same thing here.

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So an example could be the number of planes that fly over our house per hour of the day where the occurrence

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there is a plane flying over the house and the unit is time in particular ours.

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And so if we say that Lambda is equal to ten planes per hour, then we could investigate the probability

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of any number of planes flying over the house in one hour.

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But we can pick any kind of occurrence.

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And this unit doesn't have to be a unit of time.

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For instance, we could count the number of dogs that we pass per city block that we walk, and maybe

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on average we pass 1.5 dogs for every block that we walk.

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And then we could investigate the probability of passing any number of dogs, zero dogs, three dogs,

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17 dogs in any one city BLOCK.

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So again, this occurrence can be really anything we can imagine.

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And these units here can be distance, they can be area volume time.

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Any kind of occurrence per unit could in theory, be modeled by a Poisson random variable as long as

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we meet a certain set of conditions.

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And those conditions are these five.

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So again, we're looking for occurrences in an interval.

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We have to be counting a certain number of occurrences or events over some other interval unit measurement.

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That's what we described here.

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That's the first condition we have to meet.

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We have to be able to say that the mean is constant for each interval or each unit that we choose.

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So in our earlier example where we talked about planes passing over the House, this second condition

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might not hold because maybe planes only fly between certain hours of the day, maybe 6:00 am and 10:00

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PM.

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And so if we're looking at a 24 hour period, this mean is not constant across each interval.

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We only want to define a Poisson random variable when all of the intervals were investigating, as far

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as we know, at least theoretically can have the same mean.

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We also need the count of occurrences or events in each interval to be independent.

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So we need to be able to say that the number of events that we count in one interval in one unit doesn't

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affect the count we get in the other intervals.

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Our counts need to be independent across intervals, and if the count we get in one interval means that

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we're going to get a smaller or larger count in the next interval, then we don't have independent counts.

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We wouldn't meet this third condition.

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We need to have non-overlapping intervals.

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So if we're counting planes passing over the house per hour, we can't overlap our hours.

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So if we count planes from 9 to 10 a.m., then the next interval would be 10 to 11 and then 11 to 12.

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We can't count planes from 9 to 10 and then treat 930 to 1030 as another interval.

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Those intervals would overlap.

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And then finally, we have to be able to say that probability is proportional to interval size, which

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just means that if we determine that the rate at which planes pass over the House is, let's say, ten

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planes per hour, then we need to be able to.

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To expect that in any two hour interval we should see 20 planes pass over the House.

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In a three hour interval, we should see 30 planes pass over the House.

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So that probability that mean or rate lambda should scale up or scale down as we scale up or scale down

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the interval size.

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If all of these things hold, then we can model the situation with a Poisson random variable and its

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associated Poisson distribution, and we can use those two things to calculate probability of a certain

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number of K successes per unit or per interval.

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Now, when it comes to calculating probability, here's what that looks like.

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Our probability formula for the probability of K successes in one unit or one interval is lambda to

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the power of K, where K is an exponent here multiplied by e to the negative lambda.

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So negative lambda is the exponent.

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This value e here is Euler's number.

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Technically it's a constant.

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It's equal to about 2.7 approximately, but it's an infinite decimal number.

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Any calculator you use will have a button for Euler's number and then we're dividing by K factorial.

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So just to see what this looks like, let's say we're taking an example where we have a customer support

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team that receives a certain number of email support tickets per hour and they're responsible for processing

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those support tickets.

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Maybe we'll say that they receive on average four support tickets per hour such that Lambda is equal

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to four, and we then want to investigate probability around the team receiving zero support tickets

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in an hour, one support ticket in an hour to support tickets, etc..

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Well, if we first check to see that this is a Poisson process, let's bring back our criteria here.

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We are, in fact, looking for occurrences in an interval.

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We're looking for a certain number of email support tickets per hour.

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So we have email support tickets in an interval of time.

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That's the kind of situation we're modeling.

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So that checks out.

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We're saying based on data that we have, that the mean is constant in each interval.

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Maybe we have a global company and on average we do get about four support tickets per hour and that

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doesn't fluctuate significantly from one hour to the next.

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So we meet that condition.

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We'll say that the count in each interval is independent because the support tickets we receive in one

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hour don't necessarily indicate that we'll receive more or less support tickets in the next hour.

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So the interval counts are independent, Our intervals are non overlapping because we're going to look

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at each hour of the day without overlapping those hours.

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So we'll start maybe at the 8:00 hour in the morning.

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Then we'll look at the 9:00 hour, the 10:00 hour, 11:00 hour.

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We're not going to overlap our intervals.

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So we'll hold to that and we'll say that probability is proportional to interval size, because if we're

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getting about four support tickets per hour, we can expect eight support tickets every 2 hours, 12

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support tickets every 3 hours, roughly.

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So we'll say that we meet this condition as well.

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So that's kind of the idea there.

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If we say that this situation can be modeled by a Poisson random variable, then here's what probability

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looks like for this situation.

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So starting to calculate probability, what we want to look at is the probability that this customer

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support team gets zero support emails in an hour.

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Well, in that case, K is equal to zero.

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We already said that on average the support team receives Lambda equals four support tickets per hour.

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So with lambda equal to four and K equal to zero, we make substitutions into our probability formula

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and we say that we get four to the zero multiplied by E to the negative four divided by zero factorial.

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Remember that zero factorial is equal to one.

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We can use a calculator to find this value.

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If we round to four decimal places, it's approximately equal to 0.0183 or just under 2%.

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So there's a slightly less than 2% chance that the customer service team will receive exactly zero support

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emails when on average they receive four in an hour.

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So slightly less than 2% chance that they'll receive zero in any given hour.

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And then we keep everything else the same.

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We just change the value of K from 0 to 1 and then to two and then to three.

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And these are the probabilities that we get that are associated with the service team receiving exactly

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one customer support ticket in any given hour.

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Exactly.

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Two tickets.

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Exactly three tickets.

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And what you can see is that as we get closer to the lambda equals four mean value, the probability

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is rising.

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So since lambda was equal to four, the further we get away from four, the smaller probability we expect

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to see.

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So the probability that they get zero tickets is just.

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Under 2%.

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The probability that they get one ticket is a little over 7%.

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The probability they get two tickets is about 14 and one half percent.

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And the probability that they get three tickets is about 19 and a half percent.

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And then if we calculate probability for exactly four customer service emails and then five, six,

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seven, eight, we see that the probability that they get four emails is about nine and a half percent.

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And then just like we saw on this side of lambda equals four, on the other side of lambda equals four,

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we see probability start to drop off as well.

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And if we plot these probabilities in a distribution, what we see is the Poisson distribution or the

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probability mass function that's associated with this discrete random variable, this Poisson discrete

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random variable.

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And it makes sense that if Lambda is equal to four, we see these higher probability values clustered

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around that mean value of lambda equals four.

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And then on both sides of lambda equals four, the probability tapers off or decreases in both directions.

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Now realize here that on the left hand side of lambda equals four.

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We stopped at zero because of course it doesn't make sense to calculate the probability that the customer

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service team receives negative one emails in an hour.

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That's nonsensical.

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The fewest emails that they can receive is zero emails.

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So that's sort of the left hand bound of our probability distribution.

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But on the right hand side, the distribution is unbounded.

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So we could have actually continued to calculate the probability that they get nine emails, ten emails,

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11 emails, 12 emails, etc. on to infinity, and we could have found smaller and smaller and smaller

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probability values forever.

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But because those values get so insignificant at a certain point, it makes sense just to cut it off.

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So this last value that we calculated here was the probability that K is greater than eight.

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We calculated the probability that they receive exactly eight emails, and then instead of calculating

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the probability that they receive nine emails, remember that because probability always sums to one,

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we took one and then we subtracted all of these values that we calculated up through K equals eight.

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So we took one -0.0183 -0.0733 -0.1465, etc. Continuing to subtract values all the way until we got

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to this last value and we subtracted 0.0298.

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And the result of subtracting all of these values from K equals zero to K equals eight from one was

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0.0 to 1 three.

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So the remaining probability, the probability that they receive nine or more emails in any given hour,

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the probability that they receive nine emails or ten emails or 11 emails all the way on to literally

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an infinitely large number of emails, all that probability summed up together is a little more than

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0.02 or 2%.

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So there's just over a 2% chance that they receive nine or more emails.

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And we've summarized that here with this last line, and we've also summarized it here in the distribution

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to show the probability that they receive nine or more emails in any given hour.

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So those are the conditions under which we would use a Poisson random variable and how it differs from

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the binomial random variable we looked at earlier.

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And then this is the formula we use to calculate probability of K successes in any of the intervals

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that we defined, and then how to take all of those probability calculations and use them to create

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the Poisson distribution for the Poisson random variable.