Exponential family distributions | Discrete distributions | Conjugate prior distributions

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. (Wikipedia).

Bernoulli distribution
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The Bernoulli Distribution

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From playlist Probability Distributions

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The Bernoulli equation follows from a linear equation in standard form.

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This video provides an example of how to solve an Bernoulli Differential Equation. The solution is verified graphically. Library: http://mathispower4u.com

From playlist Bernoulli Differential Equations

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The Binomial Distribution

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This video provides an example of how to solve an Bernoulli Differential Equation. The solution is verified graphically. Library: http://mathispower4u.com

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Examples of finding probabilities with the Bernoulli distribution PDF. Expected value and variance, independence and links to other distributions.

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The Beta distribution is a conjugate prior for the Bernoulli. We derive the posterior distribution and the (posterior) predictive distribution under this model.

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This video provides an example of how to solve an Bernoulli Differential Equations Initial Value Problem. The solution is verified graphically. Library: http://mathispower4u.com

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From playlist Stanford CS229: Machine Learning Course | Summer 2019 (Anand Avati)

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Jacob Bernoulli | Exponential family | Beta distribution | Skewness | Conjugate prior | Statistics | Outcome (probability) | Sample mean | Boolean-valued function | Probability | Independent and identically distributed random variables | Binary entropy function | Binary decision diagram | False (logic) | Independence (probability theory) | Rademacher distribution | Bernoulli trial | Experiment | Variance | Bernoulli process | Bernoulli sampling | Bit | Geometric distribution | Random variable | Expected value | Binomial distribution | Probability theory | Kurtosis | Categorical distribution | Probability mass function