UCLA EE131A (KY) 2Gaussian (Normal) rv (1) Gaussian rv -A Gaussian rvX with two parameters (, 2)has a pdfgiven byIn (1), is called the mean, the standard deviation (sd), and 2the variance. We will shortly discuss the interpretations of these parameters. •The GaussinarvX occurs so often in theory and applications, it is also called a Normal rv•A Normal rvX with the parameters (, 2)is often denoted as X ~ N(, 2). 22-(x-μ)2σ1f(x) = , -< x < . (1)2πσe
UCLA EE131A (KY) 3Gaussian (Normal) rv (2)•In order to show f(x) given in (1) on p. 2 is a valid pdf, we show the following two properties are true:1.0 f(x). From (1), we see f(x) can not be negative. 2.Show This turn out to be non-trivial.•The “trick”is to show I2= 1. Since 0 < I, then I = 1. Let t = (x-)/and then dt= dx/.With this change of variable in f(x) of (1), then I now become-I = f(x)dx = 1 .20.50.5--22-(x-μ)-t2σ2I = (1/2π)dx/σ=(1/2π)dt .ee
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