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Binomial Distribution Binomial Distribution

In 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 Bernoulli experiment or Bernoulli 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. Know More About Exponential Distribution

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The following is an example of applying a continuity correction. Suppose one wishes to calculate Pr(X ≤ 8) for a binomial random variable X. If Y has a distribution given by the normal approximation, then Pr(X ≤ 8) is approximated by Pr(Y ≤ 8.5). The addition of 0.5 is the continuity correction; the uncorrected normal approximation gives considerably less accurate results. 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 Bernoulli 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 will of course depend on the sample. If groups of n people were sampled repeatedly and truly randomly, the proportions would follow an approximate normal distribution with mean equal to the true proportion p of agreement in the population and with standard deviation σ = (p(1 − p)/n)1/2.

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Large sample sizes n are good because the standard deviation, as a proportion of the expected value, gets smaller, which allows a more precise estimate of the unknown parameter p. Example Suppose individuals with a certain gene have a 0.70 probability of eventually contracting a certain disease. If 100 individuals with the gene participate in a lifetime study, then the distribution of the random variable describing the number of individuals who will contract the disease is distributed B(100,0.7). Note: The sampling distribution of a count variable is only well-described by the binomial distribution is cases where the population size is significantly larger than the sample size. As a general rule, the binomial distribution should not be applied to observations from a simple random sample (SRS) unless the population size is at least 10 times larger than the sample size. To find probabilities from a binomial distribution, one may either calculate them directly, use a binomial table, or use a computer. The number of sixes rolled by a single die in 20 rolls has a B(20,1/6) distribution. The probability of rolling more than 2 sixes in 20 rolls, P(X>2), is equal to 1 - P(X<2) = 1 - (P(X=0) + P(X=1) + P(X=2)). Using the MINITAB command "cdf" with subcommand "binomial n=20 p=0.166667" gives the cumulative distribution function as follows:

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Binomial Distribution