52.3.5 Geometric Random Variable

The Geometric distibution is a discrete probability distribution. It is the distribution of the number Bernoulli trials that fail before the first success.

Consider flipping a biased coin where heads occurs with probablity \(p\). Then the probability of \(k-1\) tails in a row followed by heads is given by the \({\it Geometric}(p)\) distribution.

Function: pdf_geometric (x,p) ¶

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

The pdf is

\[f(x; p) = p(1-p)^x \]

This is interpreted as the probability of \(x\) failures before the first success.

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Function: cdf_geometric (x,p) ¶

Returns the value at x of the distribution function of a \({\it Geometric}(p)\) random variable, with \(0 < p \leq 1\)

The cdf is

\[1-(1-p)^{1 + \lfloor x \rfloor} \]

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Function: quantile_geometric (q,p) ¶

Returns the q-quantile of a \({\it Geometric}(p)\) random variable, with \(0 < p <= 1\); in other words, this is the inverse of cdf_geometric. Argument q must be an element of \([0,1]\).

The probability from which the quantile is derived is defined as \(p (1 - p)^x\). This is interpreted as the probability of \(x\) failures before the first success.

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Function: mean_geometric (p) ¶

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

The mean is

\[E[X] = {1\over p} - 1 \]

The probability from which the mean is derived is defined as \(p (1 - p)^x\). This is interpreted as the probability of \(x\) failures before the first success.

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Function: var_geometric (p) ¶

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

The variance is

\[V[X] = {1-p\over p^2} \]

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Function: std_geometric (p) ¶

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

\[D[X] = {\sqrt{1-p} \over p} \]

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Function: skewness_geometric (p) ¶

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

The skewness coefficient is

\[SK[X] = {2-p \over \sqrt{1-p}} \]

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Function: kurtosis_geometric (p) ¶

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

The kurtosis coefficient is

\[KU[X] = {p^2-6p+6 \over 1-p} \]

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Function: random_geometric (p)
    random_geometric (p,n) ¶

random_geometric(p) returns one random sample from a \({\it Geometric}(p)\) distribution, with \(0 < p <= 1\).

random_geometric(p, n) returns a list of n random samples.

The algorithm is based on simulation of Bernoulli trials.

The probability from which the random sample is derived is defined as \(p (1 - p)^x\). This is interpreted as the probability of \(x\) failures before the first success.

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Categories: Package distrib · Random numbers ·