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#### 52.3.8 Negative Binomial Random Variable

The negative binomial distribution is a discrete probability distribution. Suppose we have a sequence of Bernoulli trials where each trial has two outcomes called “success” and “failure” where “success” occurs with probablity \(p\) and “failure” with probability \(1-p\). We observe the sequence until a predefined number \(r\) of sucesses have occurred. Then the number of failures seen will have a \({\it NegativeBinomial}(r,p)\) distribution.

Function: pdf_negative_binomial (x,n,p) ¶

  Returns the value at x of the probability function of a \({\it NegativeBinomial}(n,p)\) random variable, with \(0 < p \leq 1\) and \(n\) a positive number. To make use of this function, write first `load("distrib")`.

  The pdf is $$ f(x; n, p) = {x+n-1 \choose n-1} (1-p)^xp^n $$

  Categories: Package distrib ·

Function: cdf_negative_binomial (x,n,p) ¶

  Returns the value at x of the distribution function of a \({\it NegativeBinomial}(n,p)\) random variable, with \(0 < p \leq 1\) and \(n\) a positive number.

  The cdf is $$ F(x; n, p) = I_p(n,\lfloor x \rfloor + 1) $$

  where \(I_p(a,b)\) is the beta_incomplete_regularized function.

      (%i1) load ("distrib")$

      (%i2) cdf_negative_binomial(3,4,1/8);
                                    3271
      (%o2)                        ------
                                   524288

  Categories: Package distrib ·

Function: quantile_negative_binomial (q,n,p) ¶

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

  Categories: Package distrib ·

Function: mean_negative_binomial (n,p) ¶

  Returns the mean of a \({\it NegativeBinomial}(n,p)\) random variable, with \(0 < p \leq 1\) and \(n\) a positive number. To make use of this function, write first `load("distrib")`.

  The mean is $$ E[X] = {n(1-p)\over p} $$

  Categories: Package distrib ·

Function: var_negative_binomial (n,p) ¶

  Returns the variance of a \({\it NegativeBinomial}(n,p)\) random variable, with \(0 < p \leq 1\) and \(n\) a positive number. To make use of this function, write first `load("distrib")`.

  The variance is $$ V[X] = {n(1-p)\over p^2} $$

  Categories: Package distrib ·

Function: std_negative_binomial (n,p) ¶

  Returns the standard deviation of a \({\it NegativeBinomial}(n,p)\) random variable, with \(0 < p \leq 1\) and \(n\) a positive number. To make use of this function, write first `load("distrib")`.

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

  Categories: Package distrib ·

Function: skewness_negative_binomial (n,p) ¶

  Returns the skewness coefficient of a \({\it NegativeBinomial}(n,p)\) random variable, with \(0 < p \leq 1\) and \(n\) a positive number. To make use of this function, write first `load("distrib")`.

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

  Categories: Package distrib ·

Function: kurtosis_negative_binomial (n,p) ¶

  Returns the kurtosis coefficient of a \({\it NegativeBinomial}(n,p)\) random variable, with \(0 < p \leq 1\) and \(n\) a positive number. To make use of this function, write first `load("distrib")`.

  The kurtosis coefficient is $$ KU[X] = {p^2-6p+6 \over n(1-p)} $$

  Categories: Package distrib ·

Function: random_negative_binomial (n,p)
`random_negative_binomial` (n,p,m) ¶

  Returns a \({\it NegativeBinomial}(n,p)\) random variate, with \(0 < p \leq 1\) and \(n\) a positive number. Calling `random_negative_binomial` with a third argument m, a random sample of size m will be simulated.

  Algorithm described in Devroye, L. (1986) Non-Uniform Random Variate Generation. Springer Verlag, p. 480.

  To make use of this function, write first `load("distrib")`.

  Categories: Package distrib · Random numbers ·

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