Q: Moment generating and the cumulant generating function of poisson distribution?

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The MGF is exp[lambda*(e^t - 1)].

The exponential distribution and the Poisson distribution.

Yes.

Why belong exponential family for poisson distribution

Divide the total number of incidents by the total time. The result, representing the average number of incidents per unit of time, is the mean as well as the variance of the Poisson distribution.

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Lamdba (like most Greek letters in statistics) usually denotes a parameter of a distribution (usually of Poisson, gamma or exponential distributions). This will specify the entire distribution and allow for numerical analysis of the probability generating, moment generating, probability density/mass, distribution and/or cumulant functions (along with all moments), as and where these are defined.

The MGF is exp[lambda*(e^t - 1)].

Using the Taylor series expansion of the exponential function. See related links

The exponential distribution and the Poisson distribution.

The Poisson distribution. The Poisson distribution. The Poisson distribution. The Poisson distribution.

The Poisson distribution is discrete.

Yes.

Why belong exponential family for poisson distribution

we compute it by using their differences

Divide the total number of incidents by the total time. The result, representing the average number of incidents per unit of time, is the mean as well as the variance of the Poisson distribution.

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.

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.