## APPENDIX: DISTRIBUTION THEORY

DEFINITION 1 (Chi-square Distribution) Let {ZJ, і = 1, 2, . . . , n, be i. i.d. as N(0, 1). Then the distribution of X”=1Z2 is called the chi-square 9 distribution, with n degrees of freedom and denoted by Xn • 2 2 THEOREM 1 IfX~xn and T ~ Xm and if X and Y are independent, then X + Y ~ xl+m ■ THEOREM 2 If X ~ xl > then EX = n and VX = 2n. THEOREM 3 Let {X,} be i. i.d. as iV(|a, cr2), і = 1, 2, . . . , n. Define Xn = n"1 SjLiXj. Then n i= 1 2 2 Xn—1 * CT |

Proof. Define Z* = (X* — |x)/a. Then Z,- ~ N(0, 1) and |

But since (Z — Z2)/V2 ~ N{0, 1), the right-hand side of (2) is Xi by Definition 1. Therefore, the theorem is true for n = 2. Second, assume it is true for n and consider n + 1. We have

П+1 n

(3) X (Z* – Zn+l)2 = X (Zi – Zn...

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