Binomial Distribution

A random variable is said to have a binomial distribution with parameters if its distribution is

Notation: .

It can be shown that

If , then is Bernoulli distributed.

Poisson Distribution

A random variable is said to have a Poisson distribution with parameter if

Notation: .

It is well-known that

It can be proved that

that is the binomial distribution can be approximated by the Poisson distribution. The closer to zero, the better the approximation. An acceptable rule of thumb to use this procedure is és .

Geometric Distribution

A random variable is said to have geometric distribution with parameter if

Notation: .

It is easily verified that

A random variable is called modified geometric. In this case

Convolution

Definition 1.8. Let and be independent random variables with distributions , , . Then the distribution of is

is called the convolution of and , that is we calculated the distribution of the .

Example 1.1. Show that if , and are independent random variables then !

Solution:

Example 1.2. Verify if Po(), Po() and are independent random variables then !

Solution:

Example 1.3. Customers arrive at the busy supermarket according to a Poisson distribution with parameter . Each of them independently of the others becomes a buyer with probability . Find the distribution of the number of buyers.

Solution:

Let denote the number of customers and the number of buyers. By the virtue of the theorem of total probability we have

That is .