Generate the cdf from the pdf

Question

What is the process for generating the cdf from the pdf? For a simple example:

If I have pdf: $f(x) = 3x^2$

What is it's corresponding cdf, and how do I derive it?

Answer 1

If you know nothing else about probability density functions, the first thing you should know about them is this: Suppose (lower-case) $f$ is the probability density function for the distribution of a random variable $X$. Then, for every measurable set $A$ in the range of $X,$ we have $$ \Pr(X\in A) = \int_A f(u)\,du. $$

And the first thing to know about cumulative probability distribution functions is this: Suppose $F$ is the c.d.f. for the distribution of $X.$ Then $$ \Pr(X\le x) = F(x). $$

Combining these two facts, you have $$ F(x) = \Pr(X\le x) = \int_{(-\infty,\,x]} f(u)\,du = \int_{-\infty}^x f(u)\,du. $$ (Note that I wrote $(-\infty,x]$ with a round bracket on the left and a square bracket on the right.)