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Joined 1 年前
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Cake day: 2023年6月16日

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  • solution

    Say Omega = N\{0}, sigma algebra is power set of N and the probability mass function is p(n)=2^-n .

    Then A is all the even numbers, B all numbers at least 4, C all numbers at least 5.

    P(A)=sum2-n n is even = sum2-2n n is at least 1 = sum4-n n is at least 1 = 1/(1-1/4)-1=1/3

    P(B) = P(N\{1,2,3}) = 1 - 1/2 - 1/4 - 1/8 = 1/8

    P© = 1/16 similarly

    P(A and B) = P(A\{2}) = 1/3 - 1/4 = 1/12 =/= P(A)P(B) therefore not independent

    P(A and C) = P(A\{2,4}) = P(A)P© with a similar calculation and therefore independent