YLL討論網

過幾天要考試,我這幾題不會寫,請高手們救救我吧

證明以下題目

P( E1E2...En)>=P( E1)+P( E2)+...P( En)-( n-1)
用Bonferroni's inequality to n events

C [n,k]=C[i-1,k-1]

C [m+n,r]=C[n,0]C[m,r]+C[n,1]C[m,r-1]+...+C[n,r]C[m,0]

    n         n       i-1
c =Σc     ,n≧k
    k        i=k     k-1


拜託了,各位大大

不好意思上面的題目沒有打的很好,我在這裡再發一次
                           n+m      r     m     n     
1.  Prove that  C      =ΣC  C
                                    r               j=0       j       r-j


2.     
The following identity is known as Fermat’s combinatorial identity.

     n       n      n-1
C =ΣC        ,N≧K
       k        I=k        k-1
    Give a combinatorial argument (no computations are needed) to establish this identity.

Hint: Consider the set of numbers 1 through n. How many subsets of size k have I as their highest-numbered member?
3.    
From a set of n people a committee of size j is to be chosen, and from this committee a subcommittee of size i, i<xml><v> <v></v><v><v></v><v></v><v></v><v></v><v></v><v></v><v></v><v></v><v></v><v></v><v></v><v></v></v><v></v><xml><o></o></v><v><v></v></v> j, is also to be chosen.

2.     
Show that the probability that exactly one of the events E or F occurs equals P(E)+P(F)-2P(EF).
3.     
If A<v> <v></v></v>B, express the following probabilities as simply as possible:
P(A|B),  P(A|B’), P(B|A), P(B|A’)
4.     
Let A,B,C be events relating to the experiment of rolling a pair of dice.
(1)  
If  P(A|C)>P(B|C)  and  P(A|C’)>P(B|C’)  either prove that  P(A)>P(B) or give a counterexample by defining events A,B,C for which it is not true.
(2)  
If   P(A|C)>P(A|C’)  and  P(B|C’)>P(B|C’)  either prove that  P(A)>P(B) or give a counterexample by defining events A,B,C for which it is not true.
Hint: Let C be the event that the sum of a pair of dice is 10; let A be the event that the first die lands on 6; let B be the event that the second die lands on 6.
Let S be a given set. If, for some k>0, S1,S2…Sk are mutually exclusive


  k
YSi =S
i=1



, then we call the set {S1,S2,…,Sk} a partition of S. Let Tn denote the number of different partitions of {1,2,…,n}. Thus,T1=1(the only partition being S1={1}), and T2=2(the two partitions being {{1,2}},{{1},{2}}).
(1)  
Show, by computing all partitions, that T3=5,T4=15.

(2)  
Show that
                       n         n
Tn+1=1+Σ C T
                      k=1     k      k


以上,請教各位大大

    Give a combinatorial argument (no computations are needed) to establish this identity.
Hint: Consider the set of numbers 1 through n. How many subsets of size k have I as their highest-numbered member?
1.     
From a set of n people a committee of size j is to be chosen, and from this committee a subcommittee of size i, i<xml><v> <v></v><v><v></v><v></v><v></v><v></v><v></v><v></v><v></v><v></v><v></v><v></v><v></v><v></v></v><v></v><xml><o></o></v><v><v></v></v> j, is also to be chosen.
2.     
Show that the probability that exactly one of the events E or F occurs equals P(E)+P(F)-2P(EF).

3.     
If A<v> <v></v></v>B, express the following probabilities as simply as possible:
P(A|B),  P(A|B’), P(B|A), P(B|A’)
4.     
Let A,B,C be events relating to the experiment of rolling a pair of dice.

(1)  
If  P(A|C)>P(B|C)  and  P(A|C’)>P(B|C’)  either prove that  P(A)>P(B) or give a counterexample by defining events A,B,C for which it is not true.

(2)  
If   P(A|C)>P(A|C’)  and  P(B|C’)>P(B|C’)  either prove that  P(A)>P(B) or give a counterexample by defining events A,B,C for which it is not true.


Hint: Let C be the event that the sum of a pair of dice is 10; let A be the event that the first die lands on 6; let B be the event that the second die lands on 6.



這裡是中間有亂碼的地方



不好意思麻煩各位了

實在很抱歉,看來複製就會變成亂碼,訪客也沒辦法編輯,我在這裡用手打一遍

3.from a set of n people a committee of size j is to be chosen, and from this commiittee a subcom mittee of size i. i≦j, is also be chosen


4.shoe that the probability that exactly one of the event E or F occurs equals P[E]+P[F]-2P[EF]


5.If A＜B[這個符號不是尖頭的大於,是圓頭的,因為打不出來所以用大於暫時代替] express the following probabilites as simply possible
P[A∣B], P[A∣B'], P[B∣A], P[B∣A']


6.Let A,B,C be events relating to the experiment of rolling a pair od dice.

[1].If P[A∣C]＞P[B∣C] and P[A∣C']＞P[B∣C'] either prove that P[A]＞P

[2].IFP[A∣C]＞P[A∣C'] and P[B∣C] ＞P[B∣C']  either prove that P[A]＞P or give a counter example by defining events A,B,C for which it is not true.

Hint:Let C be the event that the sum of a pair of dice is 10, let A  
be the event that the first die lands on 6; let B be the event that the second die lands on 6.



7.Let S be a given set. If, for some k>0, S1,S2…Sk are mutually exclusive



前面造成閱讀困難我很抱歉