One of the best methods are described variables, and the number of data to each of the variable focus values occurring listed, this description is called the variable distribution. The ratio of the number of times each value occurs is called the probability of total number of occurrences of the value occurs. This description of the normal process, various situations arise variable can be converted to probability of each occurrence situations (random variable Z) distribution.
For a discrete random variable Z, there is an associated probability distribution function describing the probability value corresponding to each possible value Z k occur. This function is commonly referred to as Z probability mass distribution (probability mass function, abbreviated as: PMF). Probability distribution function maps each value to the probability. Corresponding to the discrete positive integer Z, can be mapped to a probability value of type float.
Two pictures taken from above - the second and third chapters "statistical probability thinking of unified mathematical programmers," a book. Pregnant women known data statistics of the number of weeks of pregnancy when a baby is born state. The picture above shows the number of weeks of pregnancy is the number of pregnant women k statistical distribution, the following figure will be mapped to the number of statistical distribution of the probability distribution.
For discrete variables z carry out the above study, the concept of probability distributions. Probability distribution described by a function called a probability mass function (PMF), which function as a parameter random variable z, z a value of 1,2, ..., k, ....
The first to introduce a probability mass function of the Poisson is a common function:
P(Z=K) = (λke-λ)/k! , k = 0,1,2,....
λ is called a parameter of this distribution, which determines the form of the distribution. For the Poisson distribution is, λ can be any positive number. As λ is increased, the probability of obtaining a large value will increase, whereas the opposite. λ is called the intensity of a Poisson distribution. The reason is widely used Poisoon function is designed λ is fantastic: Poisson distribution is equal to the expected value of the parameter λ.
If a Poisson random variable Z there is a mass distribution, we can be expressed as:
Z ~ Then (λ)
An important property of Poisson distribution is: its expected value equal to its parameters. which is:
E [Z | lambda] = l
This property is very useful, and then in order to solve practical problems, the development of the distribution of λ.
Probability mass distributions at different λ values shown below.