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negative binomial random variable (Definition)

$ X$ is a negative binomial random variable with parameters $ r$ and $ p$ if

$ f_X(x) ={r+x-1 \choose x} p^r (1-p)^x$, $ x=\{0,1,...\}$

Parameters:

$ \star$
$ r > 0$
$ \star$
$ p \in [0,1]$

Syntax:

$ X\sim NegBin(r,p)$

Notes:

  1. If $ r \in \mathbb{N}$, $ X$ represents the number of failed Bernoulli trials before the $ r$th success. Note that if $ r=1$ the variable is a geometric random variable.
  2. $ E[X] = r \frac{1-p}{p}$
  3. $ Var[X] = r \frac{1-p}{p^2}$
  4. $ M_X(t) = (\frac{p}{1 - (1-p)e^t})^r$



"negative binomial random variable" is owned by bgins. [ full author list (2) | owner history (1) ]
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Other names:  negative binomial distribution
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Cross-references: geometric random variable, variable, number, represents, syntax, parameters
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This is version 4 of negative binomial random variable, born on 2001-10-26, modified 2004-02-14.
Object id is 524, canonical name is NegativeBinomialRandomVariable.
Accessed 7322 times total.

Classification:
AMS MSC: 62E15 (Statistics :: Distribution theory :: Exact distribution theory)
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