Choosing an Appropriate Discrete Probability Distribution Function, Statistical Note 21

Choosing an appropriate probability distribution function is important to get an appropriate result. This article discusses the conditions that apply for selecting an appropriate probability distribution.

1. Conditions for Choosing an Appropriate Discrete Probability Distribution Function

Without or With Replacement: Whether all or some objects are selected one may be interested to know whether the objects are drawn without or with replacement. In selecting without replacement, the object is not replaced back once selected so that the probability of occurrence of an outcome changes from one experiment or trail to the next. This process is also referred to as sampling without replacement. Unlike, the object is replaced back after it is selected so that the object is available for following selections and the probability of occurrence does not change from one trail to next. This process is also referred to as sampling with replacement.

Two or more categories of response: An experiment can have two or more categories of response or outcomes. The number of categories also determines which discrete probability distribution function is applicable to a particular experiment or trail.

2. Appropriate Discrete Probability Distribution Functions

There are four discrete probability distribution functions that meet one or another of above conditions as presented in Diagram 1. Each one is briefly discussed below:

Diagram 1: Probability Distribution Function by Sampling technique and Number of Categories of Response

Two Category Discrete Probability Distribution With Replacement: This distribution is referred to Binomial Distribution. The probability distribution function is given as

P(X=x) = C(n,x)p^xq^n-x

Refer to my statistical note 17 for an example showing its application using the tree diagram and formula.

Two Category Discrete Probability Distribution Without Replacement: This distribution is referred to Hypergeometric Distribution. The probability distribution function is given as

P(X=x) = h(x;n,M,N) = [C(M,x) X C(N-M,n-x)]/C(N,n)

Refer to my statistical note 18 for an example showing its application using the tree diagram and formula.

Multi-Category Discrete Probability Distribution With Replacement: This distribution is referred to Multinomial Distribution. The probability distribution function is given as

P(X=x) = P(n₁,n₂,...,n_c)= [n!/(n₁! n₂! n₃! .. n_c!)](p₁ⁿ₁ p₂ⁿ₂ p₃ⁿ₃ …p_cⁿ_c)                      

Refer to my statistical note 19 for an example showing its application using the tree diagram and formula.

Multi-Category Discrete Probability Distribution Without Replacement: This distribution is referred to Multivariate Hypergeometric Distribution. The probability distribution function is given as

P(X₁=n₁, X₂=n₁, X₃=n_3… X_k=n_k) = [C(N₁,n₁) X C(N₂,n₂) X ……. C(N_k,n_k)]/C(N,n)

Refer to my statistical note 20 for an example showing its application using the tree diagram and formula.