Probability Density Function

Let X be a continuous random variable with sample space X. The probability that X take a value within the interval [a,b] is given by $$ P(a \leq X \leq b) = \int_{a}^{b} f(x) \, dx $$

where f(x) is the probability density function (pdf). Pdfs, just like pmfs must obey two conditions:

$f(x) \geq 0$ for all the x values in X.
The integral of the f(x) for all the values in X is equal to 1.

For any specific value $x_0$ in X, $P(X=x_0) = 0$ since

$$ \int_{x_0}^{x_0} f(x) \, dx = 0. $$ For example, if the waiting time of customers of a donuts shop is $$ f(x) = \begin{cases} \frac{1}{4} e^{-\frac{x}{4}}, & x \geq 0 \\ 0, & \text{otherwise} \end{cases} $$ Then the probability that the waiting time is between any two values (a,b) can be computes as $$ \int_{a}^{b} \frac{1}{4} e^{-\frac{x}{4}} \, dx. $$ In particular if we were interested in the probability that the waiting time is between two and five minutes, we could compute it as $$ P(2 < X < 5) = \int_{2}^{5} f(x) \, dx = \int_{2}^{5} \frac{1}{4} e^{-\frac{x}{4}} \, dx = 0.32 $$

$$ P(a \leq X \leq b) = P(a < X < b) = P(a \leq X < b) = P(a < X \leq b) = \int_{a}^{b} f(x) \, dx $$