Editing Geometric distribution (section)

===Fisher's Information (Geometric Distribution, Failures Before Success)===
Fisher information measures the amount of information that an observable random variable <math>X</math> carries about an unknown parameter <math>p</math>. For the geometric distribution (failures before the first success), the Fisher information with respect to <math>p</math> is given by:

:<math>I(p) = \frac{1}{p^2(1 - p)}</math>

'''Proof:'''

*The '''Likelihood Function''' for a geometric random variable <math>X</math> is:

:<math>L(p; X) = (1 - p)^X p</math>

*The '''Log-Likelihood Function''' is:

:<math>\ln L(p; X) = X \ln(1 - p) + \ln p</math>

*The Score Function (first derivative of the log-likelihood w.r.t. <math>p</math>) is:

:<math>\frac{\partial}{\partial p} \ln L(p; X) = \frac{1}{p} - \frac{X}{1 - p}</math>

*The second derivative of the log-likelihood function is:

:<math>\frac{\partial^2}{\partial p^2} \ln L(p; X) = -\frac{1}{p^2} - \frac{X}{(1 - p)^2}</math>

*'''Fisher Information''' is calculated as the negative expected value of the second derivative:

:<math>\begin{align}
I(p) &= -E\left[\frac{\partial^2}{\partial p^2} \ln L(p; X)\right] \\
     &= - \left(-\frac{1}{p^2} - \frac{1 - p}{p (1 - p)^2} \right) \\
     &= \frac{1}{p^2(1 - p)}
\end{align}</math>

Fisher information increases as <math>p</math> decreases, indicating that rarer successes provide more information about the parameter <math>p</math>.