continuing with chapter five some discrete probability distributions we are talking about the poison distribution in this video so discrete probability distributions are looking at discrete variables so we have whole numbers things that are countable in this video we're talking about poison and so for poon we are looking at the number of outcomes occurring during a given time interval or specified region so we're looking for things that we can count how many times it happened in a specific something so often this is time but it can be a specified region and we'll get to a couple examples here in a minute some of the key takeaways for Poison the number of outcomes the variable must be independent and it's also good to know that we say that the poison process has no memory and so that just means that the occurrence of one event does not affect the probability another event will occur um basically that it's independent but similarly that it has no memory there's some other things talked about in part two and part three but we're not going to really dive into them here one of the other key takeaways is that the average rate is constant and so your average rate is going to be Lambda Lambda is your rate variable so down here Lambda is the average number of outcomes per unit time so make sure you keep that per in there for Lambda we also talk about the average or the mean down here and so our mean is equal to that average rate times T often T represents time but remember it can also be a specified region so we have that the poison random variable is the number of outcomes occurring during a poison experiment or during a given interval the poison distribution is just the probability distribution of the discret poison random variable and then we have our equation down here again note that this is little F so this gives our exact probability if we want our cumulative probability we need to find our big F Lambda is the average number of outcomes per unit time T is our given time interval and the one that we are interested in in knowing more about X is the number of outcomes um and E is the weird e constant um I highly recommend knowing how to use this in your calculator with just using the value e not having to remember to type in all six of these numbers your mean and variance for the poison distribution are equal to one another they are both equal to Lambda T this also means that your Sigma is equal to the square root of Lambda T so the poison distribution looks something like this obviously these all look quite different but you can see as your me or your average increases so does your um curve begin to look more bell-shaped so here not so much here definitely not then some sample scenarios what a poison experiment might look like you might be looking at the number of downed machines in a shift the number of shipments received per day number of spam calls received during a onh hour period number of car accidents in a month number of games postponed due to rain during the baseball season so all of these in this first block here is number per time and you can see that the time intervals change we have hour day shift month baseball season down here we have the number of field mice per acre number of bacteria in a given culture number of typing errors per page and so now this is number by region and so you could also do per yard per foot per glass pane Etc whatever you are interested in looking at just know that it's per a region so there's some examples of a poon variable and now an example during a laboratory experiment the average number of radioactive particles passing through a counter in one millisecond is four so what that tells us is that the average number per time interval is four so because it's average per time that means we are looking at our Lambda so our Lambda is equal to 4 per millisecond sometimes you might get data that says the average number in two milliseconds is four then you would just change that to two milliseconds per one but in this case we have that Lambda is equal to four radioactive particles per millisecond and then it wants to know what is the probability that six particles enter the counter in any given millisecond and so this is going to be our x value our X of Interest so now all we need to do is using our little F ofx because it wants to know exactly six particles so if we take our little F ofx equal to F of six because we want to know x equal to 6 and start to plug in our variables also our T here is going to be one because it wants to know in a given millisecond so this means T is equal to one so f of6 is going to be equal to I'm actually G to break this up real quick so we're going to do f ofx is equal to P of X Lambda T so just copying down my equation E Lambda T Lambda T raised to the x / X factorial so now when we do little F of 6 equal to the probability of 6 Lambda T So 4 * 1 4 is going to be equal to e raised to -4 Lambda T is equal to 4 then 4 raised to X which is 6 ided 6 factorial can plug all of that directly into my calculator and this gives me 0.104 two so that's the final answer to my problem another way that I can do this is using my poison distribution table similar to the binomial table so this is what our poison distribution table looks like just like the binomial table we have this summation here and so that's telling us that our table is giving us our big F ofx so if we want to know our little F of six so going back to when we started talking about cumulative probabilities recall that big F of six is equal to little F of 6 Plus little F of five and so on to little F of zero recall that then big F of five is equal to little F of five plus little F of four plus yada yada all the way down to little F of zero so if what we want is this little F of six we need to subtract out big F of five so all of this needs to go away so what we want to do is take big F of 6 minus big F of 5 to get us that little F of 6 so to do that we need to find our big F of six and big F of five and so in our table we need to know our mu mu is equal to Lambda T and our R so R is going to equal our x value so for us this is going to be six and our mu is going to be 4 * 1 which is 4 so I need to find where mu is equal to 4 which is right here and then I need to find where X is equal well we are going be looking for six and five so we need to find our big F of six and big F of five down here so we need six and five so we need our five so we're going to find five drag over until we find intersect with four so we need this number and we need this number here so I need 8893 so little F of 6 is going to be equal to Big F of 6 minus big F of 5 which is equal to 0.889 3 minus 07851 which is equal to 01042 and we got the same number so doing it either way works perfectly fine depending on what you're most comfortable with and honestly what I ask for a lot of times I'll have you do either way or both way um but I will generally tell you you which way to use or use whichever way makes the most sense