Mathematical Statistics & Estimators of Parameters (Ex: Method of Moments for a Gamma Distribution) скачать в хорошем качестве

Mathematical Statistics & Estimators of Parameters (Ex: Method of Moments for a Gamma Distribution)

Applied Statistics, Class 7, Spring 2022. A statistic is a function of a random sample, which is a list of n independent random variables from the same population distribution. The sample mean and sample variance are prominent examples which are unbiased point estimators for the population mean and variance. The method of moments is used to find point estimators for the parameters α and β of a Gamma Distribution. This is used to model the Chutes and Ladders game. https://amzn.to/3rjDOoA (Probability and Statistics with Applications: A Problem Solving Text, by Asimow and Maxwell) #probability #statistics #mathematicalstatistics Links and resources =============================== 🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinn... 🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/ 🔴 Follow me on Twitter:   / billkinneymath   🔴 Follow me on Instagram:   / billkinneymath   🔴 You can support me by buying "Infinite Powers, How Calculus Reveals the Secrets of the Universe", by Steven Strogatz, or anything else you want to buy, starting from this link: https://amzn.to/3eXEmuA. 🔴 Check out my artist son Tyler Kinney's website: https://www.tylertkinney.co/ (0:00) Applied statistics should be grounded in theoretical mathematical statistics (0:45) Chutes and Ladders Mathematica simulation (5:26) Ultimate goal: relate this to a Gamma distribution (7:26) Want to make the graph of the PDF match the histogram (11:21) Populations and samples (both practically and theoretically) (13:42) Population and random sample of random variables (independent and identically distributed random variables) (15:59) Joint PDF of (X1,X2,...,Xn) is the product of n values of the PDF of X (19:38) Parameters for the population (such as mean and median) (24:06) Statistics based on the random sample (as random variables) (26:19) Sample mean statistic (estimator of the population mean μ) (27:35) Sample variance (estimator of population variance σ^2) (29:21) S^2 is an unbiased estimator for σ^2 (30:34) X bar is an unbiased estimator for μ (35:40) Proof that E[S^2]=σ^2 is in the appendix (38:02) Variance of the sample mean: Var(X bar) (43:16) Target practice analogy (46:06) The sample mean is a consistent estimator for the population mean μ (47:44) An estimator is a statistic, which is a random variable (based on a random sample of independent random variables from the same distribution) (48:48) Back to the Gamma distribution example (Chutes and Ladders simulation) (50:31) E[X]=αβ and Var(X)=αβ^2 (51:49) Can find estimators for α and β (52:26) Method of Moments estimators for α and β (parameters of a Gamma distribution) (57:23) Solve the system of nonlinear equations (59:17) Replace α and β with their estimators (1:00:31) Observed values (estimates) for Chutes and Ladders simulation (1:02:02) Observed value m2 of logical estimator for second moment E[X^2] (1:04:47) Observed estimates of α and β (1:07:23) Other topics in mathematical statistics AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.