Concept

Bernoulli distribution

In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability . Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are boolean-valued: a single bit whose value is success/yes/true/one with probability p and failure/no/false/zero with probability q. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and p would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and p would be the probability of tails). In particular, unfair coins would have The Bernoulli distribution is a special case of the binomial distribution where a single trial is conducted (so n would be 1 for such a binomial distribution). It is also a special case of the two-point distribution, for which the possible outcomes need not be 0 and 1. If is a random variable with a Bernoulli distribution, then: The probability mass function of this distribution, over possible outcomes k, is This can also be expressed as or as The Bernoulli distribution is a special case of the binomial distribution with The kurtosis goes to infinity for high and low values of but for the two-point distributions including the Bernoulli distribution have a lower excess kurtosis than any other probability distribution, namely −2. The Bernoulli distributions for form an exponential family. The maximum likelihood estimator of based on a random sample is the sample mean. The expected value of a Bernoulli random variable is This is due to the fact that for a Bernoulli distributed random variable with and we find The variance of a Bernoulli distributed is We first find From this follows With this result it is easy to prove that, for any Bernoulli distribution, its variance will have a value inside .

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Official source

https://en.wikipedia.org/wiki/Bernoulli_distribution

About this result

Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics

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Related concepts (26)

Bernoulli process

In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables Xi are identically distributed and independent. Prosaically, a Bernoulli process is a repeated coin flipping, possibly with an unfair coin (but with consistent unfairness). Every variable Xi in the sequence is associated with a Bernoulli trial or experiment.

Bernoulli trial

In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted. It is named after Jacob Bernoulli, a 17th-century Swiss mathematician, who analyzed them in his Ars Conjectandi (1713). The mathematical formalisation of the Bernoulli trial is known as the Bernoulli process.

Probability mass function

In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete probability density function. The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete.

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Use the law of Bernoulli to determine, as particular case, the fundamental law of hydrostatics for an incompressible fluid. Utilisez la loi de Bernoulli pour retrouver, comme cas particulier, la loi fondamentale de l’hydrostatique pour un fluide incompressible. The Bernoulli equation, for the particular case of static equilibrium vu2004=u20040, is written as: P(z)u2005+u2005ρgzu2004=u2004constxa0, and thus: P(z)u2004=u2004u2005−u2005ρgzu2005+u2005constxa0, Which is the equation describing the equilibrium of

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We use the siphon of the figure to empty a container of water. At the exit of the tube, the shrinking factor is 0.6. i) What is the water velocity at the exit of the tube? ii) What is the maximal height of the siphon if we want to avoid the formation of bubbles (cavitation) at the top of the siphon? The vapor pressure of water is P_(sat)u2004=u20041765 Pa. The edge of the siphon is located 2m under the bottom of the container. The flow is sufficiently weak so that we can consider the height of fluid constan