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

Categories: Package distrib ·

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

\[F(x; p) = I_{1-p}(1-\lfloor x \rfloor, \lfloor x \rfloor + 1) \]

Categories: Package distrib ·

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").

Categories: Package distrib ·

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

Categories: Package distrib ·

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

Categories: Package distrib ·

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)

Categories: Package distrib ·

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)

Categories: Package distrib ·

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

Categories: Package distrib ·

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").

Categories: Package distrib · Random numbers ·