9CommonDensities - Some Common Densities Jeffrey B Burl...

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Click to edit Master subtitle style  EE 3180 Probability and  Some Common Densities Jeffrey B. Burl 11
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 EE 3180 Probability and  Normal/Gaussian Density n The normal or Gaussian density is: n where    is the mean and  σ2  is the variance. n The Gaussian density function (and therefore all  probabilities) are fully specified by the mean and  the variance! 22 2 1 2 1 ( ) 2 x X x f x e σ πσ - - = X 2 1 2 1 ( ) 2 u X x x F x e du σ πσ - - - = 2 ( , ) N X σ
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 EE 3180 Probability and  Normal or Gaussian Density 33 -2 0 2 4 6 8 10 0 0.05 0.1 0.15 0.2 x f x ( x ) -2 0 2 4 6 8 10 0 0.5 1 x F x ( x )
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 EE 3180 Probability and  Normal RVs Continued n The integral for generating the distribution  function has no closed form solutions (same  integral for computing probabilities for Normal  RVs)! n Results tabulated or numerical solutions  given in calculator, Matlab, Mathcad, etc. n Distribution function results usually given  for standard normal RV, i. e., mean = 0 and  44
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 EE 3180 Probability and  Standard Normal RVs n The distribution function for the standard  normal RV is often tabulated: n The distribution for other normal RVs is: n Note: 55 2 1 2 1 ( ) 2 x u x e du π - - Φ = ( ) x x X F x σ - = Φ ( 29 ( 29 1 x x Φ - = - Φ
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 EE 3180 Probability and  Standard Normal RVs,  Continued n The Q function is related to standard normal  RVs and often tabulated: n The distribution for other normal RVs is: n Note: 66 2 1 2 1 ( ) 1 ( ) 2 u x Q x e du x π - = = - Φ ( ) 1 x x X F x Q σ - = - ( 29 ( 29 1 Q x Q x - = -
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 EE 3180 Probability and  Why Normal?
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