# Engineering Analysis/Probability Functions

## Probability Density Function

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

$f_X(x) = P[X = x]$

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

pdf's follow a few simple rules:

1. The pdf is always non-negative.
2. The area under the pdf curve is 1.
$\int_{-\infty}^\infty f_X(x) dx = 1$

## Cumulative Distribution Function

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:

$F_X(x) = P[X \le x]$

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

$f_X(x) = \frac{dF_X(x)}{dx}$
$F_X(x) = \int f_X(x)dx$

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

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

$P[a \le X \le b] = F_X(b) - F_X(a)$