Difference between revisions of "Bernoulli experiment"
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== Expectation and Variance == | == Expectation and Variance == | ||
| − | If the | + | If the occurrence of a success is encoded as '''1''' and the occurrence of a failure with '''0''', then a probability of a success of ''p'' (and therefore, of a failure as ''q=1-p'') leads to an expectation of |
<math>\mathbf{E}[X] = p \cdot 1 + q \cdot 0 = p</math> | <math>\mathbf{E}[X] = p \cdot 1 + q \cdot 0 = p</math> | ||
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The variance is <math>Var(X) = \mathbf{E}[(X - \mathbf{E}[X])^2] = p(1-p)^2 + q(0-p)^2 = pq </math>. | The variance is <math>Var(X) = \mathbf{E}[(X - \mathbf{E}[X])^2] = p(1-p)^2 + q(0-p)^2 = pq </math>. | ||
| − | [[ | + | The sum of ''n'' Bernoulli experiments with the same probability of success ''p'' follows a [[Binomial distribution]] ''B(n,p)''. |
| + | [[Category:Probability and Statistics]] | ||
Latest revision as of 03:08, July 13, 2016
A Bernoulli experiment (or Bernoulli trial) is the simplest non-trivial random experiment imaginable: it's an experiment of which the outcome is random and can be either of two possibilities: success and failure.
The standard example is tossing a coin.
Expectation and Variance
If the occurrence of a success is encoded as 1 and the occurrence of a failure with 0, then a probability of a success of p (and therefore, of a failure as q=1-p) leads to an expectation of
![{\displaystyle \mathbf {E} [X]=p\cdot 1+q\cdot 0=p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/661189c0ee18c6a4505b5d710a11cee8d7957065)
The variance is .
![{\displaystyle Var(X)=\mathbf {E} [(X-\mathbf {E} [X])^{2}]=p(1-p)^{2}+q(0-p)^{2}=pq}](https://wikimedia.org/api/rest_v1/media/math/render/svg/963db881693ffae85365ea6d85f52b0ec755c782)
The sum of n Bernoulli experiments with the same probability of success p follows a Binomial distribution B(n,p).