{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Probability_density_functions

# Probability_density_functions - Probability Density...

This preview shows pages 1–2. Sign up to view the full content.

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 =∞ =−∞ −∞ < < ∞ = = .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

Probability_density_functions - Probability Density...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online