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University of Toronto St. George
Statistical Sciences
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STA261H1.doc Statistical Inference R ANDOM SAMPLE Definition: Random Sample X 1, X n is called a random sample from a distribution with pxf)(or pf Px )) if 1 ,, Xn are independent and have identical distribution with pxf)(or pf Px )). It is often denoted iid (independent-identically-distributed). Note Let X 1, X n be a sample from a distribution with pdx). The joint pdf of X (X1,, X n )is f(x1,x n )= f1(x1 f n(xn) = f(x1 f (xn ). S AMPLE M EAN Definition: Sample Mean and Sample Variance n X i i=1 X1 + + X n Let X 1, X n be a random sample. The sample mean is defined by n = n = n , and the n (X i X n ) 2 i= sample variance is defined bS = n 1 . Theorem 2 If X 1, X nare iid each with a )distribution, thk1X 1+k X n n has a normal distribution 2 2 2 with mean k1++k n = k 1+k n) and variance k1 ++k n . 2 More generally, ifX1,, X n are independent and eachX ihas a N i, i ), thenk1X1++k X n nhas a 2 2 2 2 normal distribution meank1 1 ++k n n and variancek1 1 ++k n n . U SEFUL D ISTRIBUTION Theorem n Suppose X ,, X is a random sample fromN , 2 )distribution. Then X and (X X 2 )re 1 n n i n i=1 n (X i Xn 2 ) i 1 2 independent, and 2 has a n1distribution. Page 1 of 23 STA261H1.doc The t and F Distribution Two important distributions useful in statistical inference are: 1) IfW ~ N 0,1)and V ~ 2 are independent, thenW has a t-distribution. r V r U 2) U ~ r and V ~ r are independent, then r1 has a F-distribution. 1 2 V r2 T HE C ENTRAL L IMITT HEOREM Let X 1 , Xn is a random sample with finite meanand variance 2> 0 . Let n =X 1++ X n Then Sn E S n = Xn E X)n = X n N(0,1 ) var S ) 2 n n var Xn ) n S TATISTICAL M ODEL Consider a random sample X 1 , n from a distribution with pf(x). The family{f(x) } (where f(x) is pdf (oP(x)a pf), is an unknown parameter, is a parameter space) is a statistical model. We know that the distribution under investigation is in the family, but dont know which one. Based on the sample values x1, , n, we find an estimate for ; Once we find , we know the distribution. L IKELIHOOD FUNCTION Definition: The Likelihood Function Let X , , X be a random sample from a distribution with fd(x)(or pfP x )). The likelihood is 1 n defined by L: R given by L 1 , n)= c (x1, , n)= c (x1 f(xn where c > 0(or L x , x)= cP x , , x)= cP x( ) P x ) ). 1 n 1 n 1 n Definition: The Maximum Likelihood Estimate (MLE) The function : S is called the maximum likelihood estimator. (s)is called the maximum likelihood estimate of if fo ch,L (s 1 , , n) L 1 , , n). The Algorithm This suggests that in order to obtain the MLE of , we maximum the likelihood function. Since a version of the likelihood version wic =1 gives the same maximum value, we use this version. In most cases, this is done by differentiation. n 1) Write the likelihood function 1 , xn = ) f xi . ( ) i1 Page 2 of 23
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