# What are Cumulative Distribution Functions?

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

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

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.

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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, lets 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..

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