# Cumulative distribution and density functions

1. May 1, 2013

### EngnrMatt

1. The problem statement, all variables and given/known data

Let X be a random variable with probability density function:

0.048(5x-x2) IF 0 < x < 5

0 otherwise

Find the cumulative distribution function of X

a) If x ≤ 0, then F(x) =

b) If 0 < x < 5, then F(x) =

c) If x ≥ 5, then F(x) =

2. Relevant equations

Not quite sure

3. The attempt at a solution

The answer to a) is 0. The answer to c) is 1.

I am making the reasonable assumption that a) is 0 because there is no probability at that point, and that c) is 1 because after that, all probability has been "used" so to speak. However, integrating the function between 0 and 5 does not work. It seems as if my professor totally skipped over teaching us this particular type of problem. Statistics usually makes a good deal of sense to me, but this is pretty foreign.

2. May 1, 2013

### Ray Vickson

You say "integrating the function between 0 and 5 does not work". What about it does not work?

In fact, if we define f(x) = 0 for x < 0 and for x > 5, then the cumulative distribution F(z) is
$$F(z) = \int_{-\infty}^z f(x) \, dx \\ = 0 \; \text{ if } z < 0,\\ = \int_0^z (48/1000)(5x - x^2) \, dx \; \text{ if } 0 \leq z \leq 5,\\ = 1 \; \text{ if } z > 5.$$
Do the integration to see what you get.

Are you sure your course notes or textbook do not have any similar examples?