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Posted on : 28-01-2019 Posted by : Admin

If our actually observed data do not match the data expected on the basis of assumptions, we would have serious doubts about our assumptions. Such data of assumption often lead to theoretical frequency distributions also known as probability distribution. This distribution is not based on actual experimental data but on certain theoretical considerations. This may be simple two valued distribution like 3:1 as in Mendelian cross or it may be more complicated. Some of the most important probability distributions are,

- Gaussian/Normal distribution
- Binomial distribution
- Poisson distribution

Binomial and Poisson distribution apply to the discontinuous random variables and are together known as discontinuous distributions. Normal distribution applies to continuous random variables and is called as continuous distribution.

The binomial distribution describes the distribution of discrete data. James Bernoulli discovered binomial distribution in 1700. When an event can have ONLY ONE OF TWO POSSIBLE outcomes either success or failure with the probability of occurrence remaining constant then it is described as binomial distribution. And such trials are called as Bernoulli trials.

Examples of Bernoulli’s Trails are:

1) Tossing a coin (head or tail)

2) Throwing a die (even or odd number)

3) Student’s performance an exam (Pass or fail)

The probability of success is denoted by ‘p’ and that of failure by ‘q’, such that p + q = 1. The distribution can be obtained under the following experimental conditions:

1) The number of trials ‘n’ is finite.

2) The trials are independent of each other.

3) Success probability ‘p’ is constant for each trial.

4) Each trial has only one of the two possible results either success or failure.

The best exams of binomial distribution are, tossing of coins or throwing of dice or drawing cards from a pack of cards.

A discontinuous random variable x is said to follow binomial distribution if it assumes only non-negative values and its probability mass function is given by,

$\mathit{P}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{n}\mathbf{.}{\mathit{C}}_{\mathbf{x}}^{}\mathbf{}{\mathit{P}}^{\mathbf{x}}\mathbf{}{\mathit{q}}^{\mathbf{n}\mathbf{-}\mathbf{x}}$

Where,

x=0, 1…n

q= 1-p

n and p are independent constants.

n is sometimes known as degree of distribution

Here, the parameters of distribution are two independent constants. The distribution is totally determined if n and p are known. And ‘”x” refers to the number of successes.

If we consider N sets of n independent trials, then number of times we get '*x'* success is N (nCx px qn-x). It follows that the terms in the expansion of N (q + p)n gives the frequencies of the occurrences of 0,1, 2,...,x,...,n success in the N sets of independent trials.

Binomial distribution is a discrete distribution as X can take only integral values like 0, 1, 2, 3,…..n. Any variable which follows binomial distribution is known as binomial variable.

For example, a tossed coin can result in either heads or tails.

The probability of obtaining head is p=1/2 and a tail q=1/2.

So, (p + q) ^{n }=(p + q) ^{1 }

⇒1/2 + 1/2= 1

- Binomial distribution is a discrete distribution.
- Binomial distribution has two parameters n and p
- Mean of binomial distribution is np,
- Variance of binomial distribution is npq
- Standard deviation of binomial distribution √npq
- The mean is always greater than the variance
- The mode of the binomial distribution is that value of the variable which occurs with the largest probability. Binomial distribution may have either one or two modes.
- The probability of outcome of any trial is independent of the outcome of the previous trial.
- The results of each trial are characterised by two attributes namely success or failure.
- The shape and location of the binomial distribution changes as p changes for the given n. as p increases for the fixed n, the binomial distribution shifts to right.
- If two independent random variables X and Y follow binomial distribution with parameter (n1, p) and (n2, p) respectively, then their sum (X+Y) also follows Binomial distribution with parameter (n1 + n2, p)

- This distribution is mainly applied in the problems concerning:
- Number of defectives in a sample from the production line
- Number of rounds fired from the gun hitting a target
- Estimation of the reliability of the system
- Radar detection
- To know the proportion of individuals in a population having a particular character.

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