Statistical Methods

Unit 2 - Probability distributions

2.4.The Binomial distribution or Bernoulli distribution
Binomial distribution is a discrete distribution. It has great practical applications in research and industrial inspection problems. It arises when a single trial of some process or experiment can result in only one of two mutually exclusive outcomes such as male or female, with or without scales, dead or alive, whether responds or not to a given stimuli and so on.

Mathematical description of the binomial distribution can be as given below.
Suppose that the individual examined possess certain character with probability p and that does not possess it with probability 1-p=q. Then, the probability of x individuals in a sample of n individuals possessing the character is given by,

A random variable X is said to follow binomial distribution if its probability mass function is given by (1). If p and n are known, this distribution can be completely determined. Hence, p and n are called parameters of the binomial distribution. In this distribution, it is assumed that each trial results in one of two possible mutually exclusive outcomes namely ‘success’ or ‘failure’. Further, p is assumed to be constant from observation to observation and outcomes of observations are independent.

Important properties of the binomial distribution

• Mean of the binomial distribution = np
• Variance of the binomial distribution = npq
• Hence Standard deviation =
  
• As n, sample size increases the binomial distribution approaches the normal distribution.
• Various properties can be seen in the adjoining figures:

For p=q= ½ binomial distribution is symmetrical;

For p< ½ it is positively skewed;

For p> ½ it is negatively skewed;