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:

$F_{X}(x)=P[X\leq x]$

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:

$P[a\leq X\leq b]=F_{X}(b)-F_{X}(a)$