# Engineering Analysis/Probability Functions

## Probability Density Function[edit]

The **probability density function**, or **pdf** of a random variable is the function defined by:

Remember here that X is the random variable, and x is a related variable (but is not random). The subscript X on denotes that this is the pdf for the X variable.

pdf's follow a few simple rules:

- The pdf is always non-negative.
- The area under the pdf curve is 1.

## Cumulative Distribution Function[edit]

The **cumulative distribution function**, (CDF), is also known as the **Probability Distribution Function**, (PDF). to reduce confusion with the pdf of a random variable, we will use the acronym CDF to denote this function. The CDF of a random variable is the function defined by:

The CDF and the pdf of a random variable are related:

The CDF is the function corresponding to the probability that a given value *x* is less than the value of the random variable X. The CDF is a non-decreasing function, and is always non-negative.

### Example: X between two bounds[edit]

To determine whether our random variable X lies between two bounds, [a, b], we can take the CDF functions: