Geometric distribution
Method of moments

Provided they exist, the first l {\displaystyle l} moments of a probability distribution can be estimated from a sample x 1, …, x n {\displaystyle x_{1},\dotsc,x_{n}} using the formula m i = 1 n ∑ j = 1 n x j i {\displaystyle m_{i}={\frac {1}{n}}\sum _{j=1}^{n}x_{j}^{i}} where m i {\displaystyle m_{i}} is the i {\displaystyle i} th sample moment and 1 ≤ i ≤ l {\displaystyle 1\leq i\leq l}. Estimating E (X) {\displaystyle \mathrm {E} (X)} with m 1 {\displaystyle m_{1}} gives the sample mean, denoted x ¯ {\displaystyle {\bar {x}}}. Substituting this estimate in the formula for the expected value of a geometric distribution and solving for p {\displaystyle p} gives the estimators p ^ = 1 x ¯ {\displaystyle {\hat {p}}={\frac {1}{\bar {x}}}} and p ^ = 1 x ¯ + 1 {\displaystyle {\hat {p}}={\frac {1}{{\bar {x}}+1}}} when supported on N {\displaystyle \mathbb {N} } and N 0 {\displaystyle \mathbb {N} _{0}} respectively. These estimators are biased since E (1 x ¯) > 1 E (x ¯) = p {\displaystyle \mathrm {E} \left({\frac {1}{\bar {x}}}\right)>{\frac {1}{\mathrm {E} ({\bar {x}})}}=p} as a result of Jensen's inequality.
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