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The Bivariate Normal Distribution - IIT Kanpur

The Bivariate Normal DistributionThis is Section of the 1st edition (2002) of the book Introduc-tion to Probability, by D. P. Bertsekas and J. N. Tsitsiklis. Thematerial in this section was not included in the 2nd edition (2008).LetUandVbe two independent Normal random variables, and consider twonew random variablesXandYof the formX=aU+bV,Y=cU+dV,wherea, b, c, d, are some scalars. Each one of the random variablesXandYisnormal, since it is a linear function of independent Normal random variables. Furthermore, becauseXandYare linear functions of the same two independentnormal random variables, their joint PDF takes a special form, known as thebi-variate normalPDF.

2 The Bivariate Normal Distribution has a normal distribution. The reason is that if we have X = aU + bV and Y = cU +dV for some independent normal random variables U and V,then Z = s1(aU +bV)+s2(cU +dV)=(as1 +cs2)U +(bs1 +ds2)V. Thus, Z is the sum of the independent normal random variables (as1 + cs2)U and (bs1 +ds2)V, and is therefore normal.A very important …

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