Distribution function from Probability density function

Both $f$ and $F$ carry the full probability information of a continuous $X$, and either determines the other. Given the pdf $f(x)$, the cdf is built by integrating piece by piece over each interval on which $f$ has a different formula, always starting the accumulation from $-\infty$:

Worked technique (piecewise pdf). For a triangular pdf $f(x)=x-1$ on $[1,2)$, $f(x)=-x+3$ on $[2,3)$, $0$ elsewhere: for $x<1$, $F(x)=0$; for $1\le x<2$, $F(x)=\int_1^x(u-1)\,du=\tfrac{(x-1)^2}{2}$; for $2\le x<3$, $F(x)=\tfrac12+\int_2^x(3-u)\,du=1-\tfrac{(3-x)^2}{2}$; for $x\ge3$, $F(x)=1$. The same running-integral idea applies to exponential-type densities such as $f(x)=2e^{-2x}$ ($x>0$), giving $F(x)=1-e^{-2x}$ …