Cumulative Distribution Function - Continuous - Yousef's Notes
Cumulative Distribution Function - Continuous

Cumulative Distribution Function - Continuous

For a continuous random variable X the cumulative distribution function (cdf) is equally defined as $$ F(x) = P(X \leq x), $$ where now $$ P(X \leq x) = P(X < x) = \int_{-\infty}^{x} f(t) \, dt. $$

The cdf is basically the integral of the pdf.

In the donut shop example, the cdf is

$$ F(x) = \int_{-\infty}^{x} f(t) \, dt = \int_{-\infty}^{x} \frac{1}{4} e^{-\frac{t}{4}} \, dt. $$ This integral can be solved and F(x) can be calculated as $$ F(x) = 1 - e^{-\frac{x}{4}} $$