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:


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