就有 P(S1<τ1<S+h S,<T1<S1 N,=n) P(N,=n P(,=0)P(N4=1)P(N2-4=0)…P(N2=1)P(N--42=0) (r) n he-·e(2-)…入,h 定理3.12若s≤t,k≤n,则在N1=n条件下,N,的条件分布与二项分布 B(n,-)相同.而k的条件分布则是 P(T >,N1=n) P(rk≤slN,=n) P(N, =n) P(N, =J, N-N (2·D) ∑C(y n 推论3.13若s≤,k≤n则E(21M=m=4 证明用次序统计量的结果:独立同分布的U0,门]的随机变量,按小至大的第k个次 序随机变量的期望为 n+1 推论3.14Eτ,= +i 证明EN=E[E(xx|N=∑E(rn|N2=n)P(N,=n)53 就有 ( , , | ) P s1 < t1 < s1 + h1 L sn < t n < sn + hn Nt = n ( ) ( , , , ) 1 1 1 1 P N n P s s h s s h N n t n n n n t = < < + < < + = = t L t t n s h s s h h t s h e n t P N P N P N P N P N n n n l -l = = - - = = - - = = ! ( ) ( 0) ( 1) ( 0) ( 1) ( 0) 1 1 2 1 1 L t n n h t s h n s h s s h e n t e h e e h e e o h h n n n -l -l -l× -l - - -l× -l - - l × ×l× × l× × + = ! ( ) ( ) 1 ( ) ( ) 1 1 1 2 1 1 L L ,( , , 0) ! ® n h1 hn ® t n L 定理 3.１２ 若 s £ t, k £ n , 则在 Nt = n 条件下, Ns 的条件分布与二项分布 ( , ) t s B n 相同. 而 k t 的条件分布则是 å= - ÷ ø ö ç è æ ÷ - ø ö ç è æ £ = = n j k j n j j k t n t s t s P(t s | N n) C 1 (3. ９) 证明 P( s | N n) t k £ t = ( ) ( , , ) 1 P N n P s s N n t j j t n j k = £ > = = + = å t t å å = - - = = - × = = - = n j k j n j j n t n n j k s t s t s t s e n t P N j,N - N n j C ( ) (1 ) ! ( ) ( ) l l . 推论３.１３ 1 , , ( | ) + £ £ = = n kt s t k n E N n k t 若 则 t . 证明 用次序统计量的结果：独立同分布的U[0,t]的随机变量，按小至大的第k 个次 序随机变量的期望为 n + 1 kt ． 推论３.１４ t e t E t Nt l × - + l t = -l 1 证明 å ¥ = = = = = 0 [ ( | )] ( | ) ( ) n E N E E N Nt E n Nt n P Nt n t t t t t