Gaussian_or_Normal_PDF - The Gaussian or Normal Probability...

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The Gaussian or Normal Probability Density Function Author: John M. Cimbala, Penn State University Latest revision: 07 September 2007 The Gaussian or Normal Probability Density Function Gaussian or normal PDF – The Gaussian probability density function (also called the normal probability density function or simply the normal PDF ) is the vertically normalized PDF that is produced from a signal or measurement that has purely random errors . o The normal probability density function is () 2 2 2 2 2 11 ee x p 2 22 x x fx μ σ σπ −− ⎛⎞ ⎜⎟ == ⎝⎠ . o Here are some of the properties of this special distribution: ± It is symmetric about the mean. f ( x ) x ± The mean, median, and mode are all equal to , the expected value (at the peak of the distribution). Small ± Its plot is commonly called a “ bell curve because of its shape. ± The actual shape depends on the magnitude of the standard deviation. Namely, if is small, the bell will be tall and skinny, while if is large, the bell will be short and fat, as sketched. Standard normal density function – All of the Gaussian PDF cases, for any mean value and for any standard deviation, can be collapsed into one normalized curve called the standard normal density function . o This normalization is accomplished through the variable transformations introduced previously, i.e., x z = and () () f zf = x , which yields 2 /2 2 exp / 2 z fz e z ππ = . This standard normal density function is valid for any signal measurement, with any mean, and with any standard deviation, provided that the errors (deviations) are purely random . o A plot of the standard normal density function w generated in Excel, using the above equation for f ( z ). It is shown to the right. as o It turns out that the probability that variable x lies between some range x 1 and x 2 is the same as the probability that the transformed variable z lies between the corresponding range z 1 and z 2 , where z is the transformed variable defined above. In other words, ( ) ( ) 121 Px x x Pz z z <≤ = <≤ 2 where 1 1 x z = and 2 2 x z = . o Note that z is dimensionless , so there are no units to worry about, so long as the mean and the standard deviation are expressed in the same units . o Furthermore, since ( 2 1 12 x x f xd x ) , it follows that ( 2 1 z z ) f zd z . o We define A ( z ) as the area under the curve between 0 and z , i.e., the special case where z 1 = 0 in the above integral, and z 2 is simply z . In other words, A ( z ) is the probability that a measurement lies between 0 and z , or 0 z A z = d z , as illustrated on the graph below. Large
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o For convenience, integral A ( z ) is tabulated in statistics books, but it can be easily calculated to avoid the round-off error associated with looking up values in a table.
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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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Gaussian_or_Normal_PDF - The Gaussian or Normal Probability...

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