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Ch5 - Statistics 211 Prof Emanuel Parzen Chapter 5 Sampling...

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Statistics 211 Prof. Emanuel Parzen Chapter 5 Sampling Distributions, Central Limit Theorem, Normal Approximation to the Binomial This chapter will complete our set of tools of probability theory that we need to conduct statistical inference. defined as methods for learning from data the probability distribution of a random variable Y. A random sample 1 , , n Y Y of a random variable Y is usually assumed to satisfy three assumptions (whose validity needs to be checked and this is done in modern statistical practice). Random Sample Assumption 1: INDEPENDENCE. The ran- dom variables 1 , , n Y Y are independent which means that for any real numbers 1 , , n y y ( [ ] [ ] [ ] ( 29 ( 29 1 1 1 1 1 1 1 1 F , , ; , , Pr , , Pr Pr ; F ; n n n n n n n n y y Y Y Y y Y y Y y Y y F y Y y Y = = = Random Sample Assumption 2: IDENTICAL DISTRIBUTION Random variables 1 , , n Y Y have the same distribution as Y : For any real number y and 1, , j n = ( [ ] ( F ; Pr Pr F ; j j j y Y Y y Y y y Y = = = Random Sample Assumption 3: PARAMETRIC MODEL. The distribution function ( F ; | y Y true equals ( F ; | y Y θ for an un- known value of the parameter θ which we seek to learn from observed data and past experience. Introductory statistics emphasizes basic statistical meth- ods for the parameters that arise most frequently in practice: Population mean [ ] E Y μ = of Y continuous
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2 Population variance [ ] 2 VAR Y σ = of Y continuous Population probability [ ] Pr 1 p Y = = of 0 – 1 valued Y To learn the values of parameters from data we start with effi- cient estimators of the parameters. Estimators are random variables, because they are functions of the data, which become very accurate ( estimate without error) values of an unknown parameter as sample size n tends to infinity). Examples of es- timators are: Parameter Estimator μ Sample Mean ( ( M SUM / Y Y n = 2 σ Practical Sample Variance ( ( 2 SS 1 S Y n = - p Sample Proportion circumflexnosp2char , p K n K = is number of suc- cesses in n independent Bernoulli trials The variable Y that we are observing is assumed in many applications to be either (1) 0-1 valued with unknown probability p=Pr[Y=1],mean E[Y]=p, variance VAR[Y]=p(1-p) or (2) continuous obeying a Normal distribution: ( Normal , Y Z μ σ μ σ = = + where μ and σ are unknown (location and scale) parameters to be learned from the data, and ( Normal 0,1 Z = . The mean and variance of Y are [ ] [ ] [ ] [ ] 2 2 , E Y E Z VAR Y VAR Z μ σ μ σ σ = + = = = . The population quantile function of Y is ( ( ; ; Q P Y Q P Z μ σ = + where Q(P;Z) is the quantile function of a standard Normal(0,1) random variable Z. Methods for learning parameters from data use extensively facts about the distribution of sums and sample means. DISTRIBUTION OF SUMS AND SAMPLE MEANS
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3 To find the distribution of sum X+Y of the numbers (X,Y) on two fair dice, we compute the probability mass function of X+Y by the formula (that is true for X and Y discrete ran- dom variables) Pr[X+Y=k]= SUM(over j) Pr[X=j, Y=k-j] = SUM(over j) Pr[X=j] Pr[Y=k-j] For the sum X+Y of independent continuous random variables
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