Binomial DistributionBinomial DistributionIn probability theory and statistics, the binomial distribution is the discrete probability
distribution of the number of successes in a sequence of n independent yes/no experiments,
each of which yields success with probability p.
Such a success/failure experiment is also called a Bernoul i experiment or Bernoul i trial; when
n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis
for the popular binomial test of statistical significance.
The binomial distribution is frequently used to model the number of successes in a sample of
size n drawn with replacement from a population of size N.
If the sampling is carried out without replacement, the draws are not independent and so the
resulting distribution is a hypergeometric distribution, not a binomial one.
However, for N much larger than n, the binomial distribution is a good approximation, and
widely used.
In general, if the random variable K fol ows the binomial distribution with parameters n and p,
we write K ~ B(n, p).
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The probability of getting exactly k successes in n trials is given by the probability mass
function:
for k = 0, 1, 2, ..., n, where
is the binomial coefficient (hence the name of the distribution) "n choose k", also denoted C(n,
k), nCk, or nCk. The formula can be understood as fol ows: we want k successes (pk) and n -
k failures (1 - p)n - k. However, the k successes can occur anywhere among the n trials, and
there are C(n, k) different ways of distributing k successes in a sequence of n trials.
In creating reference tables for binomial distribution probability, usual y the table is filled in up
to n/2 values. This is because for k > n/2, the probability can be calculated by its complement
as
Looking at the expression (k, n, p) as a function of k, there is a k value that maximizes it.
This k value can be found by calculating
and comparing it to 1. There is always an integer M that satisfies
(k, n, p) is monotone increasing for k < M and monotone decreasing for k > M, with the
exception of the case where (n + 1)p is an integer.
In this case, there are two values for which is maximal: (n + 1)p and (n + 1)p - 1. M is the
most probable (most likely) outcome of the Bernoul i trials and is cal ed the mode. Note that
the probability of it occurring can be fairly smal .
Cumulative distribution functionThe cumulative distribution function can be expressed as : where is the "floor" under x, i.e.
the greatest integer less than or equal to x. It can also be represented in terms of the
regularized incomplete beta function, as follows:
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For k np, upper bounds for the lower tail of the distribution function can be derived. In
particular, Hoeffding's inequality yields the bound
and Chernoff's inequality can be used to derive the bound
Moreover, these bounds are reasonably tight when p = 1/2, since the following expression
holds for al k 3n/8.
This approximation, known as de Moivre-Laplace theorem, is a huge time-saver when
undertaking calculations by hand (exact calculations with large n are very onerous);
historically, it was the first use of the normal distribution, introduced in Abraham de Moivre's
book The Doctrine of Chances in 1738.
Nowadays, it can be seen as a consequence of the central limit theorem since B(n, p) is a
sum of n independent, identically distributed Bernoul i variables with parameter p. This fact is
the basis of a hypothesis test, a "proportion z-test," for the value of p using x/n, the sample
proportion and estimator of p, in a common test statistic.
For example, suppose one randomly samples n people out of a large population and ask them
whether they agree with a certain statement. The proportion of people who agree wil of
course depend on the sample.
If groups of n people were sampled repeatedly and truly randomly, the proportions would
fol ow an approximate normal distribution with mean equal to the true proportion p of
agreement in the population and with standard deviation = (p(1 - p)/n)1/2.
Large sample sizes n are good because the standard deviation, as a proportion of the
expected value, gets smal er, which allows a more precise estimate of the unknown parameter
p.
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