52.3.4 Bernoulli Random Variable

The Bernoulli distribution is a discrete probability distribution which takes on two values, 0 and 1. The value 1 occurs with probability p, and 0 occurs with probabilty 1-p.

It is equivalent to the \({\it Binomial}(1,p)\) distribution (see Binomial Random Variable)

Function: pdf_bernoulli (x,p)

Returns the value at x of the probability function of a \({\it Bernoulli}(p)\) random variable, with 0 \leq p \leq 1.

The \({\it Bernoulli}(p)\) random variable is equivalent to the \({\it Binomial}(1,p)\).

The mean is $$ f(x; p) = p^x (1-p)^{1-x} $$

(%i1) load ("distrib")$

(%i2) pdf_bernoulli(1,p);
(%o2)                           p

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Function: cdf_bernoulli (x,p)

Returns the value at x of the distribution function of a \({\it Bernoulli}(p)\) random variable, with 0 \leq p \leq 1. To make use of this function, write first load("distrib").

The cdf is $$ F(x; p) = I_{1-p}(1-\lfloor x \rfloor, \lfloor x \rfloor + 1) $$

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Function: quantile_bernoulli (q,p)

Returns the q-quantile of a \({\it Bernoulli}(p)\) random variable, with 0 \leq p \leq 1; in other words, this is the inverse of cdf_bernoulli. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

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Function: mean_bernoulli (p)

Returns the mean of a \({\it Bernoulli}(p)\) random variable, with 0 \leq p \leq 1.

The \({\it Bernoulli}(p)\) random variable is equivalent to the \({\it Binomial}(1,p)\).

The mean is $$ E[X] = p $$

(%i1) load ("distrib")$

(%i2) mean_bernoulli(p);
(%o2)                           p

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Function: var_bernoulli (p)

Returns the variance of a \({\it Bernoulli}(p)\) random variable, with 0 \leq p \leq 1.

The \({\it Bernoulli}(p)\) random variable is equivalent to the \({\it Binomial}(1,p)\).

The variance is $$ V[X] = p(1-p) $$

(%i1) load ("distrib")$

(%i2) var_bernoulli(p);
(%o2)                       (1 - p) p

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Function: std_bernoulli (p)

Returns the standard deviation of a \({\it Bernoulli}(p)\) random variable, with 0 \leq p \leq 1.

The \({\it Bernoulli}(p)\) random variable is equivalent to the \({\it Binomial}(1,p)\).

The standard deviation is $$ D[X] = \sqrt{p(1-p)} $$

(%i1) load ("distrib")$

(%i2) std_bernoulli(p);
(%o2)                    sqrt((1 - p) p)

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Function: skewness_bernoulli (p)

Returns the skewness coefficient of a \({\it Bernoulli}(p)\) random variable, with 0 \leq p \leq 1.

The \({\it Bernoulli}(p)\) random variable is equivalent to the \({\it Binomial}(1,p)\).

The skewness coefficient is $$ SK[X] = {1-2p \over \sqrt{p(1-p)}} $$

(%i1) load ("distrib")$

(%i2) skewness_bernoulli(p);
                             1 - 2 p
(%o2)                    ---------------
                         sqrt((1 - p) p)

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Function: kurtosis_bernoulli (p)

Returns the kurtosis coefficient of a \({\it Bernoulli}(p)\) random variable, with 0 \leq p \leq 1.

The \({\it Bernoulli}(p)\) random variable is equivalent to the \({\it Binomial}(1,p)\).

The kurtosis coefficient is $$ KU[X] = {1-6p(1-p) \over p(1-p)} $$

(%i1) load ("distrib")$

(%i2) kurtosis_bernoulli(p);
                         1 - 6 (1 - p) p
(%o2)                    ---------------
                            (1 - p) p

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Function: random_bernoulli (p)
    random_bernoulli (p,n)

Returns a \({\it Bernoulli}(p)\) random variate, with 0 \leq p \leq 1. Calling random_bernoulli with a second argument n, a random sample of size n will be simulated.

This is a direct application of the random built-in Maxima function.

See also random. To make use of this function, write first load("distrib").

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