Density Functions

Probability Density Function (PDF)

The probability density function can be used to measure the probability that X would fall within a specific interval. This is done by finding the area under the probability density function.

P (a \leq X \leq b) = \int_{a}^{b} f_{X} (x) d x

We can also rewrite it as such where $δ$ is a small infinitesimal value.

P (x \leq X \leq x + δ) = \int_{x}^{x + δ} f_{X} (x) d x \approx δ f_{X} (x)

And

f_{X} (x) = \frac{P (x \leq X \leq x + δ)}{δ}

Thus, $f_{X} (x)$ measures how probable X would be near x.

Joint Probability Density

For two or more variables, their joint probability density can be given as such.

P (a \leq X \leq b, c \leq Y \leq d) = \int_{a}^{b} \int_{c}^{d} x y f_{X Y} (x, y) d y d x