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Homework Help: Joint Distribution Problem

  1. May 18, 2010 #1
    Hello,

    I have a problem I need help with:

    1. The problem statement, all variables and given/known data

    There are 10 different types of coupons, each with prob 1/10 of being chosen. A total of N>=1 coupons are collected.
    Xi= 1 if type-i coupon is among the N coupons, i =1, ..., 10
    0 otherwise
    Let S=X1+X2+X3 be the number of different types out of the subset {1,2,3} contained in the collection.
    a) Find P(S=k), k=0,1,2,3


    2. Relevant equations

    Joint pmf: p(x1,x2,x3)=p(x1)*p(x2|x1)*p(x3|x1,x2)


    3. The attempt at a solution

    I know that:

    P(Xi=0)=(9/10)^N
    P(Xi=1)=1-(9/10)^N

    Therefore

    P(X1=0,X2=0,X3=0)= p(0)*p(0|0)*p(0|0,0)= (9/10)^N * (8/9)^N * (7/8)^N = (7/10)^N
    p(0|0)=(8/9)^N because if we know that there are no type-1 coupon among the N, we know that there are only 9 types.

    P(X1=0,X2=0,X3=1)= p(0)*p(0|0)*p(1|0,0)= (9/10)^N * (8/9)^N * (1-7/8)^N = (8/10)^N - (7/10)^N

    Next probability is more difficult:
    P(X1=0,X2=1,X3=1)= p(0)*p(1|0)*p(1|0,1)=

    How do I find p(1|0,1)?

    This is probability that there is a type-3 coupon among the N, given that there is no type-1 coupon, but there is type-2 coupon.
    There is no type-1 coupon, so there is only 9 types left.
    There is a type-2 coupon, so there is at least one type-2 coupon among the N, but I don't know how many... so I can only assume that there is one type-2 coupon? This would be p(1|0,1)=(1-(8/9)^(N-1)) ?

    After that P(X1=1,X2=1,X3=1) is 1 minus all the other probabilities, and my problem is complete.

    The only probability I can't find is p(1|0,1) and if anyone could help me it would be very appreciated! Thanks!!
     
  2. jcsd
  3. May 19, 2010 #2
    I think it goes something like this:
    if you have an infinite pile of coupons from which you select n, or if you put the coupon back after you look at it then:
    Notice that P(x_1)=P(x_2)=P(x_3) because they are independent. Then you are performing three bernouli trials. What kind of probability is that?
     
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