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Math Help - probability inequality invovling function

  1. #1
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    probability inequality invovling function

    Dear All.

    I have read in somewhere that if f is non decreasing function then
    p(X > \lambda) = p(f(X)>f( \lambda)), and even somewhere it is written
    p(X > \lambda) \le p(f(X)>f( \lambda)).

    My question is this how this inequality is true, and which inequality is correct?

    Can some explain in detail.

    Regards.
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  2. #2
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    Quote Originally Posted by amb03 View Post
    Dear All.

    I have read in somewhere that if f is non decreasing function then
    p(X > \lambda) = p(f(X)>f( \lambda)), and even somewhere it is written
    p(X > \lambda) \le p(f(X)>f( \lambda)).

    My question is this how this inequality is true, and which inequality is correct?

    Can some explain in detail.

    Regards.
    Neither is true. Consider f(x) = 0 for all values of x.

    Do you mean f is strictly increasing?
    In which case both are true since f(x) > f(y) iff x > y
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  3. #3
    MHF Contributor matheagle's Avatar
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    It has more to do with set theory

    If a<b and f is strictly increasing then f(a)<f(b)

    If f is strictly increasing then the two sets are equal...  \{x: x>a\} =\{x: f(x)>f(a)\}
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  4. #4
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    Thank for reply, but may you please check the following link, this inequality is stated here

    http://www.cs.cmu.edu/afs/cs/academi...ribes/lec9.pdf

    Regards
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  5. #5
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    Quote Originally Posted by matheagle View Post
    It has more to do with set theory

    If a<b and f is strictly increasing then f(a)<f(b)

    If f is strictly increasing then the two sets are equal...  \{x: x>a\} =\{x: f(x)>f(a)\}
    Would you please explain me how is it possible?. I could not understand
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