Engineering Analysis/Probability Functions

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Probability Density Function[edit]

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[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:

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[edit]

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)