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The Exponential Family of Distributions p(x)=h(x)eµ>T(x)¡A(µ) To get a normalized distribution, for any µ Z p(x)dx=e¡A(µ) Z h(x)eµ>T(x)dx=1 so eA(µ)= Z h(x)eµ>T(x)dx;. E C X g ̓W A ̂ ē Ȃǂ J Ă ܂ B @ Ō i q ̃o X P A y \ i4 A V E e j X ̉ q l ARAGNAROK ONLINE A j. I * Q ý % Ñ 0 Ô * * e Q â t J ' * % = è â s < < ¹ µ N < O K ¹ µ » < Ö o ) ( Q ¨ s Ñ % Q $ % ¹ * ¹ £ Q ² = ¹ Ä * Q · ;.

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F r T } i t C h E Β ͕ c ő S U I U A f 𗬁A Q A N W Ȃǂ ܂ B. It follows that E(s2)=V(x)−V(¯x)=σ2 − σ2 n = σ2 (n−1)n Therefore, s2 is a biased estimator of the population variance and, for an unbiased estimate, we should use σˆ2 = s2 n n−1 (xi − ¯x)2 n−1 However, s2 is still a consistent estimator, since E(s2) → σ2 as n →∞and also V(s2) → 0 The value of V(s2) depends on the form of the underlying population distribu. á ¶ Å Ä · Ñ Ñ · Ý ¹ Ò Ø Ù Í Ï µ ¹ · Ò ¹ Ó Ó ´ Ô ¹ µ Ç ¼ ¹ Í ¸ Ç 5 h = > 4 0 9 2, 1?.

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Proposition 4 All sets with µ∗(E) = 0 are µ∗ measurable Proof If µ∗(E) = 0, then for an arbitrary set A⊂Xwe have µ∗(A) ≥µ∗(A\E) = µ∗(A\E) µ∗(A∩E) {z } 0, because A∩E⊂E Let M∗ be the class of all µ∗measurable sets Theorem 5 (Carath´eodory) M∗ is a σalgebra and µ∗ M∗ →0,∞ is a measure Proof We will split the proof into several steps. ٢ Á À¿ ¾ ½ ¼ » º ¹ ¸ ¶ µ ÌË Ê É È Ç Æ Å Ä Ã Â ٣ É È Ç Æ Å Ä Ã Â Á c b a Ð Ï Î Í Ì Ë Ê o n m l k j i h g f e d. ( 0 , % E ( 6 , M % z I 6 I T ( 6 ' ( ) d$ % 8 L1 t 1 J V m l M , $ ( h C ) < 1.

9 w g é ~ d § é ù ¢ g é U Ï q D 7 u q D Z Ù é g P R b o ½ H pOT 8 ø g x X o g ÿ µ YNZJUT yK^K X U L Z µ G. E 6HNLVXL RXVH /WG ª ø ) y h V ¥ d k z 4 B V 4 j C f É C & 2 q J } 4 a ) 4 B V 4 j C Ä g V z 616. BASIC STATISTICS 1 SAMPLES,RANDOMSAMPLING ANDSAMPLESTATISTICS 11 Random Sample The random variables X1,X2,, are called a random sample of size n fromthe populationf(x)if X1,X2,, are mutuallyindependent random variablesand themar ginal probability density function of each Xi is the same function of f(x) Alternatively, X1,X2,, are called independent and identically.

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F B X G C u h A J x Z ^ i Q T r X Ə j B x ̓ e B n ̏ Q ̂ l ɑ΂ ďA J ̋@ g 傷 鎖 Ƃ s Ă ܂ B ӎu Q ̂ ɁA ` X Ɠ Ă Ƃ ̎d ł B ㋞ r f B O4F @ @ @ @ TEL @FAX. " ( , ) * ) & 2 1 0 / < ;. References ABeck,FirstOrder Methods in Optimization (17),chapter6 PLCombettesandJChPesquet,Proximal splitting methods in signal processing,inFixedPoint Algorithms for Inverse Problems in Science and Engineering (11) NParikhandSBoyd,Proximal algorithms (13) Theproximalmapping 624.

Probability 2 Notes 5 Conditional expectations E(XjY) as random variables Conditional expectations were discussed in lectures (see also the second part of Notes 3) The. DoorTrim_DecoFlash_J65 ARCAT, Inc DecoFlash.