Bernoulli distribution

Bernoulli distribution
Probability mass function
Funzione di densità di una variabile casuale normale

Three examples of Bernoulli distribution:

   and
   and
   and
Parameters


Support
PMF
CDF
Mean
Median
Mode
Variance
MAD
Skewness
Excess kurtosis
Entropy
MGF
CF
PGF
Fisher information

In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,[1] 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. [2]

  1. ^ Uspensky, James Victor (1937). Introduction to Mathematical Probability. New York: McGraw-Hill. p. 45. OCLC 996937.
  2. ^ Dekking, Frederik; Kraaikamp, Cornelis; Lopuhaä, Hendrik; Meester, Ludolf (9 October 2010). A Modern Introduction to Probability and Statistics (1 ed.). Springer London. pp. 43–48. ISBN 9781849969529.