Binomial Distribution

History

The desire to calculate probabilities in games of chance led to the first studies of binomial distribution. Essentially, mathematically inclined gamblers wanted to calculate their probability of winning on a certain number of dice rolls. In 1713, Jakob Bernoulli, a Swiss [Page 124]mathematician, published a proof that determined that the probability of Χ equaling a specified number, x, in n trials was equal to the xth term in the binomial expansion of the expression (p + q)n, thus creating the binomial distribution.

The binomial distribution was used in 1936 to publish evidence of possible scientific chicanery by Gregor Mendel in the famous 1866 pea genetics experiments. Ronald Fisher noted that the reported laws of inheritance in peas would dictate that the number of certain colors of peas would have a binomial distribution, and the results reported by Mendel should have a probability of only about .1.

Applications for Statistical Inference

Any experiment using the binomial distribution has two assumptions: identical trials and independent trials. Identical trials is the assumption that p and q take on the same probability value across n number of trials. The compound binomial distribution does not assume identical trials. Independent trials is the assumption that subsequent trials are not affected by previous outcomes, so in order for the binomial distribution to be used, sampling with replacement must occur.

When the random variable Χ has parameters n and p in the binomial distribution, it is mathematically represented as Χ~B(n, p). The probability that Χ is equal to a certain number, x, where x = 0, 1, 2, . . . , n, is given by the probability mass function:

$$f\left(x\right)=P\left(C=x\right)=C_x^n{p}^x{q}^{n-x},$$

where $C_x^n$ is a mathematical combination, called the binomial coefficient, given by the equation:

$$C_x^n=\frac{n!}{x!\left(n-x\right)!}.$$

The cumulative distribution function for the binomial distribution is simply the sum of the probability mass function results for all applicable values. For example, the cumulative distribution function for the probability that Χ is less than or equal to a certain number, x, where x = 0, 1, 2, . . . , n, is given

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