作者mangogogo (mangogo)
看板Statistics
标题Re: [机率] 多个随机变数的问题...
时间Mon Oct 9 20:14:15 2006
※ 引述《homal (\冏/)》之铭言：
: X1 , X2 ... 为 independent identically distributed
: continuous random variable (独立,相同,连续,的随机变数)
: we say: a record occurs at time n
: if Xn > max (X1,X2...Xn-1)
: 即 Xn 比 X1,X2 ...Xn-1 都来的大.
: show:
: a) P{a record occurs at time n} = 1/n
=P{X_(n)=Xn}=(n-1)!/n!=1/n
: b) E[number of records by time n] = Σ(1/i) (i=1 ~ n)
Let Y_i=1 if a record occurs at time i
0 for all i
E[Y_i]=P{Y_i=1}=P{a record occurs at time i}=P{X_(i)=Xi}=1/i
E[number of records by time n]=E[Y_1+Y_2+...+Y_n]=Σ(1/i)
: c) Var(number of records by time n) = Σ(i-1)/(i^2) (i=1 ~ n)
E[(Y_i)^2]=P{Y_i=1}=1/i
Var[Y_i]=E[(Y_i)^2]-(E[Y_i])^2=1/i-(1/i)^2=(i-1)/(i^2)
Var[number of records by time n]=Var[Y_1+Y_2+...+Y_n] (independent)
=ΣVar[Y_i]=Σ(i-1)/(i^2)
: d) if N = min{n:n>1 and a record occurs at time n}
: 证明 E[N] = ∞
: 想问一下， 像(b)的 number of records by time n ,
那个 by 是 「乘」的意思吗???NO
: 另外 (b)在求的东西是什麽呢?!?
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※ 编辑: mangogogo 来自: 218.170.125.16 (10/09 20:14)
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