19  Geometric and Negative Binomial Distribution

19.1 Geometric Random Variable

Suppose that independent trials, each having probability \(p\) of being a success, are performed until a success occurs.

If we let \(X\) be the number of trials required until the first success, then \(X\) is said to be a geometric random variable with parameter \(p\). Its probability mass function is given by: \[p(x) = P\{X = x\} = (1-p)^{x-1}p \qquad x=1,2,...\]

The geometric random variable \(Y\) is sometimes defined as the number of failures required until the first success. Then, its probability mass function is: \[p(y) = P\{Y = y\} = (1-p)^{y}p \qquad y=0,1,...\]

Remark: \(Y = X - 1\)

Proposition: The geometric random variable has memoryless property, i.e. \[\begin{align}(X - c | X > c) &\sim X \\ (Y - c | Y \ge c) &\sim Y\end{align}\]

Properties: * \(\text E[X] = \frac{1}{p}\) and \(\text E[Y] = \frac{1}{p}-1\) * \(\text{Var}(X) = \text{Var}(Y) = \frac{1-p}{p^2}\) * \(\varphi_X(t) = \frac{p}{e^{it}-(1-p)}\) and \(\varphi_Y(t) = \frac{p}{1-(1-p)e^{it}}\) * \(G_X(z) = \frac{p}{z-(1-p)}\) and \(G_Y(z) = \frac{p}{1-(1-p)z}\) ## Negative Binomial Random Variable Suppose that independent trials, each having probability \(p\) of being a success, are performed until \(r\) success occurs.

If we let \(X\) be the number of trials required until the \(r\)-th success, then \(X\) is said to be a negative binomial random variable with parameters \((r,p)\). Its probability mass function is given by: \[p(x) = P\{X = x\} = {x-1\choose r-1}(1-p)^{x-r}p^r \qquad x=r,r+1,...\] The negative binomial random variable \(Y\) is sometimes defined as the number of failures required until the \(r\)-th success. Then, its probability mass function is: \[p(y) = P\{Y = y\} = {y-r-1\choose r-1}(1-p)^{y}p^r \qquad y=0,1,...\]Remark: \(Y = X - r\).

Proposition: Let \(X_i \sim NB(r_i, p)\) be independent negative binomial random variables. Then \[X_1 + ... + X_m \sim NB\bigg(\sum_{i=1}^m r_i, p\bigg)\] The geometric random variable is the special case of the negative binomial random variable where \(r=1\).

Properties: * \(\text E[X] = \frac{r}{p}\) and \(\text E[Y] = \frac{r}{p} - r\) * \(\text{Var}(X) = \text{Var}(Y) = \frac{r(1-p)}{p^2}\) * \(\varphi_X(t) = \big(\frac{p}{e^{it}-(1-p)}\big)^r\) and \(\varphi_Y(t) = \big(\frac{p}{1-(1-p)e^{it}}\big)^r\) * \(G_X(z) = \big(\frac{p}{z-(1-p)}\big)^r\) and \(G_Y(z) = \big(\frac{p}{1-(1-p)z}\big)^r\)