https://wikimd.com/index.php?action=history&feed=atom&title=Binomial_distributionBinomial distribution - Revision history2026-04-20T05:47:13ZRevision history for this page on the wikiMediaWiki 1.44.2https://wikimd.com/index.php?title=Binomial_distribution&diff=5633039&oldid=prevPrab: CSV import2024-04-19T20:09:50Z<p>CSV import</p>
<p><b>New page</b></p><div>[[File:Binomial_distribution_pmf.svg|Binomial distribution pmf|thumb]] [[File:Binomial_distribution_cdf.svg|Binomial distribution cdf|thumb|left]] [[File:Pascal's_triangle;_binomial_distribution.svg|Pascal's triangle; binomial distribution|thumb|left]] [[File:Binomial_Distribution.svg|Binomial Distribution|thumb]] '''Binomial Distribution''' is a [[probability distribution]] that summarizes the likelihood that a value will take on one of two independent values under a given set of parameters or assumptions. The concept is widely used in [[statistics]], [[probability theory]], and various fields that involve decision making under uncertainty, such as [[finance]], [[healthcare]], and [[engineering]].<br />
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==Definition==<br />
The binomial distribution is defined by two parameters: \(n\) and \(p\). Here, \(n\) represents the number of trials, and \(p\) represents the probability of success on an individual trial. The random variable \(X\), which follows a binomial distribution, represents the number of successes in \(n\) trials.<br />
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The probability mass function (PMF) of a binomial distribution is given by:<br />
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\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]<br />
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where \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \), \(k\) is the number of successes, \(n-k\) is the number of failures, \(p\) is the probability of success, and \(1-p\) is the probability of failure.<br />
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==Characteristics==<br />
===Mean===<br />
The mean, or expected value, of a binomial distribution is given by \( \mu = np \).<br />
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===Variance===<br />
The variance of a binomial distribution is given by \( \sigma^2 = np(1-p) \).<br />
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===Standard Deviation===<br />
The standard deviation is the square root of the variance, \( \sigma = \sqrt{np(1-p)} \).<br />
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==Applications==<br />
Binomial distributions are used in a variety of fields to model binary outcomes. For example, in [[healthcare]], it can be used to model the probability of a certain number of patients recovering from a disease out of a total number of cases. In [[quality control]], it can model the number of defective items in a batch of products.<br />
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==Examples==<br />
1. Coin Toss: If a fair coin is tossed 10 times, the probability of getting exactly 6 heads can be calculated using the binomial distribution with \(n=10\) and \(p=0.5\).<br />
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2. Quality Control: If a factory produces items with a 2% defect rate, the probability of finding exactly 5 defective items in a sample of 100 can be calculated using the binomial distribution with \(n=100\) and \(p=0.02\).<br />
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==Limitations==<br />
While the binomial distribution is widely applicable, it has limitations. It assumes a fixed number of trials, a constant probability of success, and independent trials. When these assumptions do not hold, other distributions, such as the [[Poisson distribution]] or the [[negative binomial distribution]], may be more appropriate.<br />
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[[Category:Probability Distributions]]<br />
[[Category:Statistics]]<br />
[[Category:Mathematical and Quantitative Methods]]<br />
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