Probability_density_functions - Probability Density...

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Probability Density Functions Author: John M. Cimbala, Penn State University Latest revision: 04 September 2007 Probability Density Functions Probability density function – In simple terms, a probability density function ( PDF ) is constructed by drawing a smooth curve fit through the vertically normalized histogram as sketched. You can think of a PDF as the smooth limit of a vertically normalized histogram if there were millions of measurements and a huge number of bins. o The main difference between a histogram and a PDF is that a histogram involves discrete data (individual bins or classes), whereas a PDF involves continuous data (a smooth curve). x f ( x ) x 1 x 2 x 3 ... 0.02 0.03 0 0.01 o Mathematically, f ( x ) is defined as () 22 ii i dx dx Px x x fx dx ⎛⎞ −< + ⎜⎟ ⎝⎠ = , where dx dx + represents the probability that variable x lies in the given range , and f ( x ) is the probability density function (PDF) . In other words, for the given infinitesimal range of width dx between x i dx /2 and x i + dx /2, the integral under the PDF curve is the probability that a measurement lies within that range , as sketched. x f ( x ) x i + dx /2 0.02 0.03 0 0.01 x i /2 x i + o As shown in the sketch, this probability is equal to the area (shaded blue region) under the f ( x ) curve – i.e., the integral under the PDF over the specified infinitesimal range of width dx . o The usefulness of the PDF is as follows: Suppose one chooses a range of variable x , say between a and b . The probability that a measurement l between a and b is simply the integ under the PDF curve between a and b , as sketched, where we define the probability as ies ral ( xb xa Pa x b f xd x = = <≤ = x f ( x ) b 0.02 0.03 0 0.01 a P ( a < x b ) ) o If a and b + , the probability must equal 1 (100%), i.e., ( ) 1 x x f x d x =∞ =−∞ −∞ < < ∞ = = .
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This note was uploaded on 04/05/2008 for the course ME 345 taught by Professor Staff during the Spring '08 term at Pennsylvania State University, University Park.

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Probability_density_functions - Probability Density...

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