| Version 7 |
Version 6 |
| A \textbf{geometric random variable} with parameter $p\in(0,1]$ is one whose density distribution function is given by |
A \textbf{geometric random variable} with parameter $p\in(0,1]$ is one whose density distribution function is given by |
| \begin{equation*} |
\begin{equation*} |
| f_X(x) = p(1-p)^x,\qquad x=0,1,2,\dotsc |
f_X(x) = p(1-p)^x,\qquad x=0,1,2,\dotsc |
| \end{equation*} |
\end{equation*} |
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| This is denoted by $X\sim Geo(p)$ |
This is denoted by $X\sim Geo(p)$ |
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| Notes: |
Notes: |
| \begin{enumerate} |
\begin{enumerate} |
| \item A standard application of geometric random variables is where $X$ represents the number of failed Bernoulli trials before the first success. |
\item A standard application of geometric random variables is where $X$ represents the number of failed Bernoulli trials before the first success. |
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\item The expected value of a geometric random variable is given by $E[X] = \frac{1}{p}$, and the variance by $Var[X] = \frac{1-p}{p^2}$
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\item The expected value of a geometric random variable is given by $E[X] = \frac{1-p}{p}$, and the variance by $Var[X] = \frac{1-p}{p^2}$
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| \item The moment generating function of a geometric random variable is given by $M_X(t) = \frac{p}{1 - (1-p)e^t}$. |
\item The moment generating function of a geometric random variable is given by $M_X(t) = \frac{p}{1 - (1-p)e^t}$. |
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| \end{enumerate} |
\end{enumerate} |
