Risk theory Decision-making
under uncertainty

Description

A random variable X possesses Bernoulli distribution with parameters a,b,p, where a < b and 0 < p < 1, if it takes only values a è b, and

P(X = a) = 1 - p, P(X = b) = p.

Denote this variable by Ba,b,p, and Bp - standard Bernoulli variable with parameters a = 0, b = 1.

The following picture presents graphs of the probability function (bound to the left axis) and CDF (bound to the right axis) for B 0,2,0.3. Probability function and CDF

Characteristics

The next table contains formulae for calculation of characteristics of Bernoulli distribution B a,b,p.

Probability function
CDF
Expectation a(1-p) + bp
Variance p(1-p)(b-a)2
Asymmetry (1-2p) / [p(1-p)]1/2
Kurtosis - 6 + 1 / p(1-p)
median Does not make sense
Mode Exercise for readers: calculate the mode

Simulation

To get values of a random variable Ba,b,p one can use the following method.

  1. Get U with uniform distribution on [0,1].
  2. If U < p, then set Ba,b,p = b, else set Ba,b,p = a.

Simulation of uniform distribution on [0,1] is described here.

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