Inclusion probability

The sampling rate, and selection, inclusion or inclusion probability, rare sample weights ( engl. inclusion probability ) indicates the probability with which access one or more elements of a population in a sample.

As inclusion probability first order the probability denotes that the ith element of the population is included in a sample of size. The inclusion probability 2nd order analog ( with ) with the get the probability the i-th and j -th element in a sample of size.

In a total or simple random sampling, the inclusion probabilities can specify directly. For more complex sampling design effects occur. Here also, not every element has the same probability to get into the sample.

Calculation of inclusion probabilities

The inclusion probability of first order can be for a full or simple random sample calculated using the hypergeometric distribution:

Since there is only an i-th element in the population, and either M = 1, it is preferred (k = 1 ) or not ( k = 0):

Accordingly, the following applies:

Analogously, the inclusion probability of 2nd order for an unrestricted random sample charge; here M = 2 and k = 2:

Example

The population consists of four elements: { w1, w2, w3, w4 }. We consider three samples of size n = 2, namely { w1, w3 }, { w2, w4 } and { w3, w4 }. In an unrestricted random sample, there would be total possible samples; that is, only the above three random samples, it is not an unrestricted random sample.

The probability for each of the samples is just 1/3 and the inclusion probabilities result to be

• Sampling theory