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CONCEPTUAL TOOLS By: Neil E. Cotter P ROBABILITY LINEAR COMBINATIONS RV'S Normal/gaussian dist EXAMPLE 1 EX: Find the probability density function of the average value of 12 independent standard gaussian random variables. SOL'N: We want to find the probability density function for Y: 12 1 Y = ∑ Xi 12 i=1 where € Xi ~ n(0,1) are independent standard gaussian random variables We rewrite the expression for Y to emphasize that it is a linear combination of independent random variables. Y= 1 1 X1 + K + X12 12 12 Thus, we have the following mean and variance: € € € µY = 1 1 1 µ X1 + K + µ X12 = 12 ⋅ ⋅ 0 = 0 12 12 12 1 2 1 2 1 2 1 σY2 = σ 2X + K + σ 2X = 12 ⋅ ⋅1 = 1 12 12 12 12 12 Also, the probability density function (pdf) for a linear combination of independent gaussian random variables is a gaussian random variable. Thus, the pdf for Y is the following gaussian distribution: fY (y) = NOTE: € 1 2πσY2 e −(y−µY ) 2 / 2σ Y2 The pdf for a linear combination of independent gaussian random variables, Xi, is gaussian even when the Xi have differing means and variances.