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What is the difference between a binomial distribution and a poisson distribution?

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Anonymous Posted

What is the difference between a binomial distribution and a poisson distribution?

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Oliver Morgan

Both the Poisson and binomial distributions are used to model situations in which we have a large number of trials, each of which is capable of producing a binary output – henceforth referred to as true and false outcomes, although the actual result could be anything from ‘Red vs Blue’ to ‘Conviction vs Acquittal’. Indeed, in many cases we an use a poisson distribution to approximate a binomial distribution, as we often do when dealing with especially large numbers of trials. However, in general a Poisson distribution will be used to model situations in which we know the mean or average likelihood of a given event occurring, whilst a binomial distribution will be used when we know the exact probability of an event occurring. If an event can occur a finite number of times, then you will use a binomial distribution, whereas if an event can theoretically occur any number of times up to infinity, you will use a Poisson distribution. It is easiest to demonstrate this difference with an example.

If, for example, we knew that our car broke down on average twice a year, and we wished to calculate the probability that it will break down three times in the next two years, we would use a Poisson distribution. This is because we:
a)    Are dealing with an average.
b)    Are dealing with a potentially infinite number of events – there is no reason why our car could not, with very low probability, break down 100 times in the next two years.

If, on the other hand, we were noting the colour of cars as they pass us on the road, and we wished to know the likelihood that at least 3 of the next 10 cars to pass us will be red, we would use a binomial distribution. This is because we:
a)    Are dealing with an exact probability. (From monitoring the cars to this point, we have a numerical probability that a given car will be red.)
b)    Are dealing with a definitely finite number of events. The maximum number of red cars is 10. We are looking at 10 trials, exactly.

At the end of the day, it can often be a judgement call as to which one is more appropriate for a given situation, but I hope that this helped a little. Feel free to message me if you need any further clarification.

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kiranya Talla

Binomial Distribution : It is used when the experiment has only two outcomes. 

Poisson Distribution: It is as limiting case of binomial distribution with n→∞, p→0,and mean that is m=np remaining finite. Therefore the ‘r’ successes in a large number of ‘n’ trails with small number of probability success p in each trail

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Dheeraj Kumar

Binomial distribution tells the probability of no of success and failures in n no of trails where as poisson distribution tells or represents a discrete frequency distribution which gives the probability of a number of independent events occurring in a fixed time

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