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Strictly increasing cdf

  1. Nov 9, 2011 #1
    Let's say we have a cumulative distribution function (cdf) G and random numbers v1 and v2.

    The definition of strict increasing function is: v1 < v2 => G(v1) < G(v2).

    In a statistics book, the author writes:

    "...but with the additional assumption that the cdf G is a strictly increasing function. That is, v1 < v2 <=> G(v1) < G(v2)".

    a) He writes the definition with an equivalence (<=>) and not an implication (=>). Could someone explain why? Does in fact the definition also imply that its is an equivalence?

    b) The authors definition: "v1 < v2 <=> G(v1) < G(v2)" must imply that:
    v1 = v2 <=> G(v1) = G(v2). Correct?

    Any help is very appreciated!
    Last edited: Nov 9, 2011
  2. jcsd
  3. Nov 9, 2011 #2
    I think it is because "false implies true" is true, so he wanted to avoid that by using <->. Or maybe it is just an insignificant detail. I think that your 2 conclusion is correct and is easy to prove.
  4. Nov 9, 2011 #3
    The reverse implication (<=) is true because G is a cdf.
  5. Nov 9, 2011 #4


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    Science Advisor

    As bpet said, CDF's have this property.
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