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Q: What is the difference between Geometric and a Poisson probability distribution?

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The Poisson distribution is discrete.

It can be.

Poisson and Binomial both the distribution are used for defining discrete events.You can tell that Poisson distribution is a subset of Binomial distribution. Binomial is the most preliminary distribution to encounter probability and statistical problems. On the other hand when any event occurs with a fixed time interval and having a fixed average rate then it is Poisson distribution.

The Poisson distribution is characterised by a rate (over time or space) of an event occurring. In a binomial distribution the probability is that of a single event (outcome) occurring in a repeated set of trials.

Independent events with a constant probability of occurrence over a fixed interval of time (or space).

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The Poisson distribution is discrete.

A poisson process is a non-deterministic process where events occur continuously and independently of each other. An example of a poisson process is the radioactive decay of radionuclides. A poisson distribution is a discrete probability distribution that represents the probability of events (having a poisson process) occurring in a certain period of time.

Why belong exponential family for poisson distribution

Poisson distribution or geometric distribution

It can be.

Poisson and Binomial both the distribution are used for defining discrete events.You can tell that Poisson distribution is a subset of Binomial distribution. Binomial is the most preliminary distribution to encounter probability and statistical problems. On the other hand when any event occurs with a fixed time interval and having a fixed average rate then it is Poisson distribution.

The normal distribution is a continuous probability distribution that describes the distribution of real-valued random variables that are distributed around some mean value.The Poisson distribution is a discrete probability distribution that describes the distribution of the number of events that occur within repeated fixed time intervals, where the mean frequency is a known value, and each interval is independent of the prior interval(s)/event(s).

Poisson distribution shows the probability of a given number of events occurring in a fixed interval of time. Example; if average of 5 cars are passing through in 1 minute. probability of 4 cars passing can be calculated by using Poisson distribution. Exponential distribution shows the probability of waiting times between occurrences of events. If we use the same example; probability of a car coming in next 40 seconds can be calculated by using exponential distribution. -Poisson : probability of x times occurrence -Exponential : probability of waiting times between events.

For the binomial, it is independent trials and a constant probability of success in each trial.For the Poisson, it is that the probability of an event occurring in an interval (time or space) being constant and independent.

The binomial distribution is a discrete probability distribution. The number of possible outcomes depends on the number of possible successes in a given trial. For the Poisson distribution there are Infinitely many.

we compute it by using their differences

It depends on the parameter - the mean of the distribution.