→ NiceKitty:謝謝大大喔^^ 01/14 23:31
→ goshfju:推改符號 XD 01/15 00:11
※ 引述《NiceKitty (想要有個依靠><)》之銘言：
: 不好意思再請教各位大大一題><
: Consider n independent trials,each of which results in any of the
: outcomes i, i=1,2,3 with respective probabilities P1,p2,p3
: 3
: ΣPi=1.Let Ni denote the number of trials that result in outcome i,
: i=1
: and show that Cov(N1,N2).
: Also explain why it is intuitive that this covariance is negative.
n=N1+N2+N3,1=P1+P2+P3 =>N3=n-N1-N2,P3=1-P1-P2
∴(N1,N2)~Trinomial(n,P1,P2)
n! n1 n2 n-n1-n2
f(n1,n2)=------------------------(P1) (P2) (1-P1-P2) , n1=0,1,...,n
(n1)!*(n2)!*(n-n1-n2)! n2=0,1,...,n-n1
則
N1~Binomial(n,P1) , N2~Binomial(n,P2) , N1與N2不獨立
E(N1)=nP1 , E(N2)=nP2 , V(N1)=nP1(1-P1) , V(N2)=nP2(1-P2) , Cov(N1,N2)=-nP1P2
最後二行是直接寫結論出來，推導可以自己推推看
一般研所的機率或統計的書都會介紹到可以去翻翻看
另外N1跟N2改成X跟Y比較不會搞混，這是個人看法
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