Strictly increasing cdf

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  • #1
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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!
 
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Answers and Replies

  • #2
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!
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.
 
  • #3
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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!
The reverse implication (<=) is true because G is a cdf.
 
  • #4
chiro
Science Advisor
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As bpet said, CDF's have this property.
 

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