Introduction to Binomial Distribution

Definition

A more valuable probability density function with many applications is the binomial distribution. This distribution will compute probabilities for any binomial process. A binomial process, often called a Bernoulli process after the first person to fully develop its properties, is any case where there are only two possible outcomes in any one trial, called successes and failures. It gets its name from the binary number system where all numbers are reduced to either 1’s or 0’s, which is the basis for computer technology and CD music recordings.

Binomial Formula

$b\left ( x \right )=\binom{n}{r}p^{x}q^{n-x}$

where b(x) is the probability of X successes in n trials when the probability of a success in ANY ONE TRIAL is p. And of course q=(1-p) and is the probability of a failure in any one trial.

Characteristics of a binomial

There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials.
The random variable, , number of successes, is discrete.
There are only two possible outcomes, called “success” and “failure,” for each trial. The letter p denotes the probability of a success on any one trial, and q denotes the probability of a failure on any one trial. p + q = 1.
The n trials are independent and are repeated using identical conditions. Think of this as drawing WITH replacement. Because the n trials are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability, p, of a success and probability, q, of a failure remain the same. For example, randomly guessing at a true-false statistics question has only two outcomes. If a success is guessing correctly, then a failure is guessing incorrectly. Suppose Joe always guesses correctly on any statistics true-false question with a probability p = 0.6. Then, q = 0.4. This means that for every true-false statistics question Joe answers, his probability of success (p = 0.6) and his probability of failure (q = 0.4) remain the same.

Key Points

The binomial distribution is a probability distribution that summaries the likelihood that a value will take one of two independent values under a given set of parameters or assumptions.
The underlying assumptions of the binomial distribution are that there is only one outcome for each trial, that each trial has the same probability of success, and that each trial is mutually exclusive or independent of each other.
The binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as the normal distribution.

Negative binomial

The negative binomial distribution is a discrete probability distribution which estimates the number of failures (f) before the sth success in a number of independent trials where each trial has the same probability (p) of success.The pmf of the NBD is:

$p_{x\binom{k+x-1}{x}} p^{k}q^{x},0<k,0<p<1,q=1-p,x=0,1,2,....$

It has the probability mass function:

$p\left ( X=s \right )=\binom{s+r-1}{s}p^{s}\left ( 1-p \right )^{r}$

s = number of success;
r = number of failures before the experiment stops;
p = is the probability of success.

It is a discrete probability distribution which has two input parameters (s and p) and there are two forms of the negative binomial:

negative binomial (s, p);
negative binomial (s + 1, p).

The first form is taken when we know that we stop sampling after s successes and provides the number of failures before we achieve s successes, the second form is taken when the ordering is not known, but we know that there are just s successes, hence it provides the number of failures before we observe s successes. An example is if you play the lottery scratch card game, where one in five has a cash prize and you want to be assured of five wining cards. There are two scenarios:

You buy one ticket and scratch it and continue this cycle until you have the required number of winning tickets (represented by negative binomial (s, p)). You stop buying tickets when you have successfully obtained the five winning tickets, so we know the final ticket is a winning ticket.

You go into a shop and bulk buy the tickets (represented by negative binomial (s+1,p)) and the order of the winning tickets is not known (i.e. the last ticket may not necessarily be a winning one).