Binomial Poisson and Normal Distribution

Binomial Distribution:

A Bernoulli trial is one in which there are just two possible outcomes: true or false or success or failure. The Bernoulli trial sample space is given by S={success,failure}. The random variable that counts the number of successes in a single Bernoulli trial is called a Bernoulli random variable. For instance, if X is a random variable, it provides, based on given values, the number of successful attempts.

If X is a Bernoulli random variable, we may write,
X ~ BER (p)
Where ~ is read as “is distributed as” and BER stands for Bernoulli.

The experiment must be a Bernoulli Process in order to satisfy the requirement for a binomial random variable.
An experiment needs to possess the following three characteristics in order to be a Bernoulli process:

• In the experiment, there are only two outcomes that are mutually exclusive and collectively exhaustive: success and failure. 
• In subsequent trials of the experiment, the odds of occurrence of these events do not change.
• The trials' results stand alone from one another.

A binomial random variable is one that counts the number of successful trials across numerous independent, identical Bernoulli trials, and a binomial distribution is the probability distribution of a binomial random variable.

Binomial Probability Function:

Using the two parameters n and p, we can characterize the distribution of a binomial random variable as follows:

X ~ B (n, p)
to indicate that X is Binomially distributed with n number of independent trials and p probability of success is each trial. The letter B stands for binomial.

Let us analyze the probability that the number of successes X in the n trials is exactly x (obviously number of failures are n-x) i.e. X = x and x = 0,1,2,…………. n; as n trials are made, at the best all n can be successes. Now we know that there are nCx ways of getting x successes out of n trials. We also observe that each of these nCx possibilities has px(1-p)n-x probability of occurrence corresponding to x successes and (n-x) failures.

Therefore, P(X = x) = nCx p x (1-p)n-x for x = 0,1,2,………, n

This equation is the Binomial probability formula. If we denote the probability of failure as q then the Binomial probability formula is

P(X = x) = nCx p x qn-x for x = 0,1,2,………, n

Important formulas: 

• The mean of the binomial distribution = n*π
• The mean is the ‘expected’ value of X
• The variance of the binomial distribution = n π(1-π)
• The standard deviation of the binomial distribution = \(\sqrt{nπ(1-π)}\)

Poisson Distribution:

The probability distribution for discrete random variables is called the Poisson distribution. Simeon D. Poisson (1781–1840), a French mathematician, is remembered in the name of this formula, which is used to estimate the likelihood of a given number of events occurring during a given period of time or area. Events that occur over time or space include:

• Number of emergency room beds occupied in a 24-hour period.
• Number of cells in a given volume of fluid.
• Number of calls made to a help line each hour.

There are numerous Poisson distributions, just like the binomial distribution. The average rate at which the event happens determines each Poisson distribution. The rate is denoted by the Greek letter "L" (lambda, which means "rate") The Poisson distribution has just one parameter. The likelihood that X specific occurrences occur in the specified space or time is given as follows:

p(X)=\frac{λ^xe^{-λ}}{x!}

Where, x is the number of successes λis the mean of the Poisson distribution and e = 2.71828 (the base of natural logarithms).

The random variable X counts the number of successes in Poisson Process. A Poisson process corresponds to a Bernoulli process under the following conditions: ¾ the number of trials n, is infinitely large i.e. n → ∝¾ the constant probability of success p, for each trial is infinitely small i.e. p → 0 (obviously q → 1) ¾ np = λ is finite.

A Poisson distribution has an identical mean and variance, which are both equal to λ. The square root of λ is the Poisson distribution's standard deviation.

Normal Distribution:

It is preferable to use the normal distribution if or is greater than 10. A specific type of these symmetric distributions that may be defined by two parameters is the Normal distribution.

• µ = The mean of the distribution
• σ = The standard deviation of the distribution

Changing the values of µ and σ alter the positions and shapes of the distributions.

If X is Normally distributed with mean µ and standard deviation σ, we write X∼N(µ, σ2) µ and σ are the parameters of the distribution.