Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Geometric distribution
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
=== 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}}
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)