What are the four properties that must be present in order to use binomial distribution? What are four properties that must be present in order to use the Poisson distribution?

1) The binomial distribution could be used if these rules are met:

Rule 1: If there are only two mutually exclusive outcomes for a discrete random variable (i. e., success or failure).

Rule 2: If there is a fixed number of repeated trials (i. e., successive tests with no outcome excluded).

Rule 3: If each trial is an independent event (meaning the result of one trial doesn't affect the results of subsequent trials).

Rule 4: If the probability of success for each trial is fixed (i. e., the probability of obtaining a successful outcome is the same for all trials).

2) The Poisson distribution could be used when the properties below are met:

Rule 1: k is the number of times an event that occurs in an interval and can take values 0, 1, 2, ...

Rule 2: The occurrence of one event does not affect the probability and a second event will occur, that is why, events occur independently.

Rule 3: The average rate at which events occur is constant.

Rule 4: Two events are not able to occur at exactly the same instant but, instead, at each very small sub-interval exactly one event either occurs or does not occur.

Explanation:

1) Binomial distribution

In probability theory and statistical sciences, the binomial distribution applies to n trials of two categories. The category of interest of the researcher is called success. In each such experiment, it is independently known that the probability of success (yes = 1) is p (and that the probability of failure is 1 - p because only two categories of results are possible). The discrete probability distribution of the number of successes obtained in this series of independent n number of trials is defined as the binomial distribution. A binomial distribution is defined exactly by only two parameters, namely n and p. If a random variable X denotes a binomial distribution as a mathematical notation, it is expressed as follows:

X ~ B (n, p)

In this way, each trial that results in success / failure is also referred to as the Bernoulli trial. If n = 1, this binomial distribution, B (1, p), is actually the same as a Bernoulli distribution. Binomial distribution reveals the basic theory for the used binomial test which is very popular in inferential statistical analysis and practical problem solving efforts.

The binomial distribution could be used if these rules are met:

Rule 1: If there are only two mutually exclusive outcomes for a discrete random variable (i. e., success or failure).

Rule 2: If there is a fixed number of repeated trials (i. e., successive tests with no outcome excluded).

Rule 3: If each trial is an independent event (meaning the result of one trial doesn't affect the results of subsequent trials).

Rule 4: If the probability of success for each trial is fixed.

2) Poisson Distribution

Poisson distribution is a discrete probability distribution in probability theory and statistical sciences and expresses the probability of the number of occurrences in a fixed time unit range. It is assumed that the average number of events occurring in this time interval is known and that the time difference between any event and the event immediately following it is independent of previous time differences. The Poisson distribution is often applied to problems with fixed fixed time units, but can also be successfully applied to other unitary spaced problems (i. e., problems involving unit distance, area or volume). The random variable on which the Poisson distribution is generally focused is a countable event; this event occurs discrete in a fixed length (usually time) interval, and the number of events observed in that interval is a random variable for the Poisson distribution. The expected value of the number of events occurring in this fixed interval (the average number of occurrence) is constant as λ, and this average value is proportional to the interval length. If an average of 5 events occur within each 4-minute interval, an average of 10 events occurs at a fixed 8-minute interval. The probability that any non-negative integer k occurs is expressed as follows:

f (k, λ) = (λ ^ k * e ^ ( - λ)) / k!

-e, base of the natural logarithm (e = 2.71828 ...);

-k, the number of occurrences of the event being given by the function;

-k!, factorial for k

-λ is the expected value of the number of occurrences in the given fixed range; a positive real number.

The Poisson distribution could be used when the properties below are met:

1) k is the number of times an event that occurs in an interval and can take values 0, 1, 2, ...

2) The occurrence of one event does not affect the probability and a second event will occur, that is why, events occur independently.

3) The average rate at which events occur is constant.

4) Two events are not able to occur at exactly the same instant but, instead, at each very small sub-interval exactly one event either occurs or does not occur.