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The Bivariate Normal Distribution

Let U and V be two independent normal random variables, and consider two new random variables X and Y of the form X = aU + bV, Y = cU + dV, where a, b, c, d, are some scalars. Each one of the random variables X and Y is normal, since it is a linear function of independent normal random variables.† Furthermore, because X and Y are linear functions of the same two independent normal random variables, their joint PDF takes a special form, known as the bivariate normal PDF. The bivariate normal PDF has several useful and elegant properties and, for this reason, it is a commonly employed model. In this section, we derive many such properties, both qualitative and analytical, culminating in a closed-form expression for the joint PDF. To keep the discussion simple, we restrict ourselves to the case where X and Y have zero mean.