Question

What is the difference between a binomial, hypergeometric and Poisson probability distribution?

Answer 1

We know that in binomial probability distribution we calculate the probability of a success or a failure in an experiment that is repeated multiple times. We know that hypergeometric distribution is a probability distribution which can just be considered as an expansion of binomial distribution. And in Poisson distribution, we find the probability of a given number of events happening in a fixed duration of time.

Complete step-by-step answer:
We know that a probability distribution is a table or equation that relates each outcome of a statistical experiment with its probability of occurrence. So, we can say that the probability distribution gives information about the possible outcome of any random event.
We know that binomial probability distribution is simply the probability of a success or a failure outcome in an experiment that is repeated multiple times. This type of distribution has only two possible outcomes. We are also aware that the number of trials is fixed in such distribution. We must also keep in mind that such an experiment is conducted with replacement. We also have the information about probability of success, which is a constant for each trial.
We know that hypergeometric distribution is a probability distribution which can be considered as an expansion of binomial distribution. In this type of distribution, the number of trials is fixed. But, we know very well that in this distribution, no replacement is done. Also, we know that the probability for each trial changes in this distribution.
Through Poisson distribution, we can find the probability of a given number of events happening in a fixed interval of time. In this distribution, the number of trials is unknown and we represent it using a random variable. We are aware that replacement takes place in this type of distribution. And, instead of the actual probability, we know about the average probability.
These are the basic differences in binomial, hypergeometric and Poisson distribution.

Note: We know that binomial distribution is a very good approximation of the hypergeometric distribution as long as we are sampling 5% or less of our sample space. We must take care that the events are independent in binomial and Poisson, but not in hypergeometric.