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 $\mathit{G}\mathit{e}\mathit{o}\mathit{m}\mathit{e}\mathit{t}\mathit{r}\mathit{i}\mathit{c}(p)$ distribution.

Function: pdf_geometric (x,p) ¶

Returns the value at x of the probability function of a $\mathit{G}\mathit{e}\mathit{o}\mathit{m}\mathit{e}\mathit{t}\mathit{r}\mathit{i}\mathit{c}(p)$ random variable, with $0<p\le 1$

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 $\mathit{G}\mathit{e}\mathit{o}\mathit{m}\mathit{e}\mathit{t}\mathit{r}\mathit{i}\mathit{c}(p)$ random variable, with $0<p\le 1$

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 $\mathit{G}\mathit{e}\mathit{o}\mathit{m}\mathit{e}\mathit{t}\mathit{r}\mathit{i}\mathit{c}(p)$ 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 $\mathit{G}\mathit{e}\mathit{o}\mathit{m}\mathit{e}\mathit{t}\mathit{r}\mathit{i}\mathit{c}(p)$ random variable, with $0<p\le 1$.

The mean is

$$E[X]=\frac{1}{p}-1$$

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 $\mathit{G}\mathit{e}\mathit{o}\mathit{m}\mathit{e}\mathit{t}\mathit{r}\mathit{i}\mathit{c}(p)$ random variable, with $0<p\le 1$.

The variance is

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

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Function: std_geometric (p) ¶

Returns the standard deviation of a $\mathit{G}\mathit{e}\mathit{o}\mathit{m}\mathit{e}\mathit{t}\mathit{r}\mathit{i}\mathit{c}(p)$ random variable, with $0<p\le 1$.

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

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Function: skewness_geometric (p) ¶

Returns the skewness coefficient of a $\mathit{G}\mathit{e}\mathit{o}\mathit{m}\mathit{e}\mathit{t}\mathit{r}\mathit{i}\mathit{c}(p)$ random variable, with $0<p\le 1$.

The skewness coefficient is

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

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Function: kurtosis_geometric (p) ¶

Returns the kurtosis coefficient of a geometric random variable $\mathit{G}\mathit{e}\mathit{o}\mathit{m}\mathit{e}\mathit{t}\mathit{r}\mathit{i}\mathit{c}(p)$ , with $0<p\le 1$.

The kurtosis coefficient is

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

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

random_geometric(p) returns one random sample from a $\mathit{G}\mathit{e}\mathit{o}\mathit{m}\mathit{e}\mathit{t}\mathit{r}\mathit{i}\mathit{c}(p)$ 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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