Question

The probability destiny function is where statistics and probability come together. While there are several different kinds of discrete probability functions (or PDF's), three in particular are most commonly used. These are the binomial, Poisson and hypergeometric. What are the characteristics of each? Where and how are they used? Have you ever seen or even used any of these?

Answer 1

Binomial distribution

The characteristics of Binomial distribution:

1. There are a fixed number of trials i.e n is finite.
2. There are only two possible outcomes, called “success” and “failure,” for each trial.
3. p denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial.
4. The n trials are independent and are repeated using identical conditions.
5. It has two parameters n and p
6. The probability mass function is , where x=1, 2,....n
7. Mean of binomial distribution is np and variance npq .

Binomial distribution is used when n (no. of trials) is finite and event are independent of each other. There are only two possible outcomes success and failure. The outcomes are mutually exclusive and collectively exhaustive.

Poisson distribution:

The characteristics of Poisson distribution:

1. The event or a success are counted in whole numbers.
2. The probability of having a success in a time interval is independent of any of its previous occurrence.
3. The average frequency of successes in a unit time interval is known. (λ in the above case).
4. The probability of more than one success in a unit time is very low. (since n → ∞ and p → 0 anyway).
5. It has on one parameter i.e .
6. The Probability mass function is where x=0,1,...∞
7. Mean = Variance =

Poisson distribution is used when n (no. of trials) is infinite or unknown. There are only two possible outcomes success and failure. It is used in rarer of rarest case or when since n → ∞ and p → 0.

Hypergeometric distribution

The characteristics of Hypergeometric distribution:

1. Trails are dependent
2. There are also only two outcome “success” and “failure,”
3. The probability of success is not same on each trial without replacement.
4. Events are not independent
5. Population is finite.
6. The Probability mass function is where  N is the population size, K is the number of success states in the population, n is the number of draws (i.e. quantity drawn in each trial) and x is the number of observed successes.
7. There are three  parameters of N, K, and n

Hypergeometric distribution is used to find the probability of success in n draws without replacement. The successive trials is dependent on the previous trial.

We have used these distributions in various situation like:

1. If a coin is tossed 10 times then a binomial distribution can be used to determine the probability of 4 heads.
2. Poisson distribution can be used to find the probability of 5 phone call in one hour.
3. Hypergeometric distribution can be used to find out the probability that out of 5 items chosen from a set of 24 items (out of which 5 are defective) 1 is defective.