Methods 03 [Schreibgeschützt]
Parameter Estimation Numero, pondere et mensura Deus omnia condidit Population, Random Variable, Sample population random variable X sample x a "realisation" of X (observation) all male US army recruits (population ) the first 10 male conscripts entering the US army recruitment office in Concord NH on 11-5-2004 ( sample ) the BMI of a male US army recruit to be chosen at a randomly selected time from a randomly selected recruitment office ( random variable ) Some Conventions Random variables are denoted by capital letters . Possible values or actual observations ('realisations') are denoted by small letters . P(X=x) probability that the BMI of a randomly drawn recruit is equal to x P(X>18) probability that a randomly drawn recruit is not underweight P(24<X<30) probability that a randomly drawn recruit is overweight, but not obese Estimation from Samples Basic Idea Population Sample X x ,...,x random variable Sampling 1 n θ: parameter realisations Conclusion Data Collection ˆθ ˆθ Inference (x1 ,...,xn ) estimate estimator Estimation from Samples Basic Idea Parameter Observations Estimator θ θ) x1,...,x n (x1 ,..., xn ) π π = 0,0,1,1,0,1,... ˆ k / n probability proportion µ µ = 1.23,4.81,7.55,... ˆ x expected value sample mean σ2 σ2 = 2 12.4,19.6,20.4,... ˆ s variance sample variance Coin Tossing π: probability of head 0.30 π = k = 6 = ˆ 0.6 0.25 n 10 0.20 0.15 X: number of heads in 10 tosses 0.10 X follows a Bin( π,10) distribution 0.05 = = ⋅ π6 ⋅ − π 4 0.00 Pπ (X 6) 210 (1 ) 0.0 0.2 0.4π 0.6 0.8 1.0 Likelihood and Probability The Likelihood of a parameter, given the data, equals the Probability of the data, given the parameter.
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