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.
Discrete
The triangular, uniform, binomial, Poisson, geometric, exponential and Gaussian distributions are some that can be so defined. In fact, the Poisson and exponential need only the mean.
Yes.
Why belong exponential family for poisson distribution
In parametric statistical analysis we always have some probability distributions such as Normal, Binomial, Poisson uniform etc.In statistics we always work with data. So Probability distribution means "from which distribution the data are?
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.
Discrete
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.
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.
The Poisson distribution with parameter np will be a good approximation for the binomial distribution with parameters n and p when n is large and p is small. For more details See related link below
If X and Y are i.i.d Poisson variables with lambda1 and lambda2 then, P (X = x | X + Y = n) ~ Bin(n, p) where p = lambda1 / lambda1 + lambda2
The Poisson distribution. The Poisson distribution. The Poisson distribution. The Poisson distribution.
If this is the only information that you have then you must use the Poisson distribution.
The triangular, uniform, binomial, Poisson, geometric, exponential and Gaussian distributions are some that can be so defined. In fact, the Poisson and exponential need only the mean.
The Poisson distribution is discrete.
Yes.
Why belong exponential family for poisson distribution