What are Cumulative Distribution Functions?

[xeon123]

As I understand, Cumulative Distribution Functions gives the probability that a value X be lower than what is estimated. Is this right?

 

[Avatrin]

It tells you the probability of getting at most x (ie P(X ≤ x)). If the function is continuous it will be:
∫ p(x)dx from -inf to x

 

[xeon123]

If I've graph where the Y axis represents the CDF, and the YY values goes from 0 to 1, what it means?

Imagine a graph that as the XX values grow, the YY values also grow.

 

[Avatrin]

The Y-axis on the CDF graph represents P(X ≤ x)
The CDF always goes from 0 to 1 (because you cannot have less than 0% chance of getting something and no more than 100%)

Well, let's take the classic coin toss as a discrete example:

You flip the coin twice. You count the number of heads (i.e. heads = 1, tails = 0)
You have four possibilities:

0+0 = 0
0+1 = 1
1+0 = 1
1+1 = 2

I guess you know that the probability of getting any of these specific outcomes is equal to 1/4. The probability of getting P(X ≤ 0) = 1/4 (i.e. only 0+0)

The probability of getting P(X ≤ 1) is 3/4. Why? Because three of the possibilities above are equal to or less than 1 (0+0, 0+1 and 1+0).

P(X ≤ 2) = 1 because all of the possibilities above are equal to or less than 2..

 

[blue_raver22]

Yes, that is a correct understanding of Cumulative Distribution Functions (CDFs). CDFs are a way to describe the probability distribution of a random variable, such as a measurement or observation. They show the probability that the random variable will take on a value less than or equal to a given value. This can be useful in analyzing data and making predictions in various fields of science, such as statistics, physics, and economics. Essentially, CDFs help us understand the likelihood of different outcomes and can be used to make informed decisions based on that information.