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.

   $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{f}_X(x)dx=1$

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:

$f_X(x)=\frac{d{F}_X(x)}{dx}$

$F_X(x)=\int f_X(x)dx$

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:

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