Editing Geometric distribution (section)

=== Method of moments ===
Provided they exist, the first <math>l</math> moments of a probability distribution can be estimated from a sample <math>x_1, \dotsc, x_n</math> using the formula<math display="block">m_i = \frac{1}{n} \sum_{j=1}^n x^i_j</math>where <math>m_i</math> is the <math>i</math>th sample moment and <math>1 \leq i \leq l</math>.<ref name=":5">{{Cite book |last1=Evans |first1=Michael |url=https://www.utstat.toronto.edu/mikevans/jeffrosenthal/ |title=Probability and Statistics: The Science of Uncertainty |last2=Rosenthal |first2=Jeffrey |year=2023 |isbn=978-1429224628 |edition=2nd |pages= |publisher=Macmillan Learning |language=en}}</ref>{{Rp|pages=349–350}} Estimating <math>\mathrm{E}(X)</math> with <math>m_1</math> gives the [[sample mean]], denoted <math>
\bar{x}
</math>. Substituting this estimate in the formula for the expected value of a geometric distribution and solving for <math>
p
</math> gives the estimators <math>
\hat{p} = \frac{1}{\bar{x}}
</math> and <math>
\hat{p} = \frac{1}{\bar{x}+1}
</math> when supported on <math>\mathbb{N}</math> and <math>\mathbb{N}_0</math> respectively. These estimators are [[Biased estimator|biased]] since <math>\mathrm{E}\left(\frac{1}{\bar{x}}\right) > \frac{1}{\mathrm{E}(\bar{x})} = p</math> as a result of [[Jensen's inequality]].<ref name=":3">{{Cite book |last1=Held |first1=Leonhard |url=https://link.springer.com/10.1007/978-3-662-60792-3 |title=Likelihood and Bayesian Inference: With Applications in Biology and Medicine |last2=Sabanés Bové |first2=Daniel |date=2020 |publisher=Springer Berlin Heidelberg |isbn=978-3-662-60791-6 |series=Statistics for Biology and Health |location=Berlin, Heidelberg |language=en |doi=10.1007/978-3-662-60792-3}}</ref>{{Rp|pages=53–54}}