# Cumulative distributed function example

1. May 10, 2012

### xeon123

I was looking to a video about cumulative distribution function () and he show the following function:

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | 1/4, 0 \leq x \leq1 \\ f(x) =<(x^3)/5, 1 \leq x \leq 2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |0, otherwise.$

At minute 8:45, he presents the cumulative distribution as:

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | 0, x \leq 0 \\ F(x) = < \frac{1}{4}x, 0 \leq x \leq 1 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | \frac{1}{20}(x^4+4), 1 \leq x \leq 2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | 1, \ x \geq 2$

I don't understand why F(x) is 1 for $x \geq 2$, if f(x) is 0, otherwise. Why?

BTW, I hope that that my functions are legibles, because I don't know how to put big curly brackets.

Last edited by a moderator: Sep 25, 2014
2. May 10, 2012

### Stephen Tashi

Look at a simpler example. Suppose f(x) = 1/2 when x = 1 or x = 2 and f(x) = 0 otherwise. The value of the cumulative distribution F(x) would be 1 at x = 3 because F(3) gives the probability that x is equal or less than 3. The condition that x is equal or less than 3 includes the cases x = 1 and x = 2.

3. May 11, 2012

### xeon123

I understand what you said, but the probability of happening 3 is 0, because it's not defined in f(x). For me, F(3) should never be defined.

4. May 11, 2012

### Stephen Tashi

Do you mean "should" in some moral or religious sense? Mathematics would only care about you opinion if you could show some logical contradiction in the standard definition of cumulative distribution function.

It is defined in the domain of f(x). f(3) = 0. That's part of the "f(x) = 0 otherwise" clause.

Last edited: May 12, 2012
5. May 11, 2012

### HallsofIvy

You seem to be thinking that "F(3)" is the probability that x is equal to 3. That is not the case. F(X) is the probability that x is less than or equal to 3. Since, by the definition of f(x), x must be less than or equal to 2, x therefore must be less than or equal to 3. F(x)= 1 for any number larger than or equal to 2.

If f(x) is the "probability density function" then $F(X)= \int_{-\infty}^X f(x)dx$ is the probability that x is less than or equal to X.