Continuous Probability Distributions

Probability distribution

A probability distribution is a function that gives the probability of occurrence of a random event. It can be either discrete or continuous, depending on the domain. We can formally define it as:

$$\mathcal P: \mathcal A\to \mathcal R$$

Here $\mathcal A$ can determine the distribution as either discrete or continuous as being a discrete or continuous set, respectively.

There are a couple of useful ways to describe a probability distribution:

• Cumulative Distribution Function (CDF)
• Probability Density Function (PDF)

Cumulative Distribution Function (CDF)

The CDF of a distribution is the value it will take for $x$ less than (or equal to) the given value (generalized as $X$). Formally,

$$F_X(x) = {P}(X\leq x) = \int_{-\infty}^x f_X(u) \, du$$

Note: Since probabilities are always non-negative, CDF is a non-decreasing function.

Probability Density Function (PDF)

More often, we are interested in calculating the value for a particular $x$ rather than the whole limit. Here we can use the Probability Density Function (PDF). This can be calculated by taking the derivative of the CDF:

$$f_X(x) = \frac{d}{dx} F_X(x)$$ ...