Why is the cdf G considered a strictly increasing function?

This is because a CDF is a function mapping from a set to the interval [0, 1], and as such, it must preserve the ordering of its inputs. Therefore, if v1 < v2, then G(v1) < G(v2), and vice versa. This is why the author uses an equivalence (<=>) instead of an implication (=>) in the definition of a strict increasing function.
  • #1
PAHV
8
0
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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  • #2
PAHV said:
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
PAHV said:
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
As bpet said, CDF's have this property.
 
  • #5


a) The author likely uses an equivalence instead of an implication to emphasize the fact that the relationship between v1 and v2 and G(v1) and G(v2) is bidirectional. In other words, v1 < v2 implies G(v1) < G(v2) and vice versa. This is important because it means that the cdf G is a one-to-one function, meaning that each input (v1) has a unique output (G(v1)). This is necessary for a cdf to accurately represent the probability distribution of a random variable.

b) Yes, the author's definition does imply that v1 = v2 if and only if G(v1) = G(v2). This is because if v1 = v2, then the inequality v1 < v2 is not true, and therefore G(v1) < G(v2) cannot be true either. Similarly, if G(v1) = G(v2), then the inequality G(v1) < G(v2) is not true, and therefore v1 < v2 cannot be true either. This reinforces the idea that the cdf G is a one-to-one function.
 

1. What is a strictly increasing cdf?

A strictly increasing cdf (cumulative distribution function) is a mathematical function that maps the probability of a random variable being less than or equal to a given value. It is strictly increasing if the probability increases as the value of the random variable increases.

2. How is a strictly increasing cdf different from a non-increasing cdf?

A strictly increasing cdf only increases as the value of the random variable increases, while a non-increasing cdf can either increase or stay constant as the value increases. This means that a strictly increasing cdf has a steeper slope compared to a non-increasing cdf.

3. What are some examples of variables with strictly increasing cdfs?

Examples of variables with strictly increasing cdfs include time, height, weight, and income. These variables have a natural order and their probabilities increase as the value increases.

4. How is a strictly increasing cdf useful in statistics?

A strictly increasing cdf is useful in statistics because it can help calculate the probability of a random variable falling within a certain range or above a certain value. It is also used in hypothesis testing and determining confidence intervals.

5. Can a cdf be both strictly increasing and strictly decreasing?

No, a cdf cannot be both strictly increasing and strictly decreasing. This is because a strictly increasing cdf only increases as the value of the random variable increases, while a strictly decreasing cdf only decreases as the value increases. These two behaviors are contradictory and cannot occur simultaneously.

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