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132 3. Multivariate Random Variables
of the distribution. A random variable (X , X ) having the pdf given by (3.6.1)
1
2
is denoted by N (µ , µ , , ρ). The pdf f(x , x ) from (3.6.1) is centered
2 1 2 1 2
around the point (µ , µ ) in the x , x plane. The bivariate normal pdf given by
1
2
2
1
(3.6.1) has been plotted in the Figures 3.6.1-3.6.2.
The Figure 3.6.2 (a) appears more concentrated around its center than
its counterpart in the Figure 3.6.1 (a). This is due to the fact that we have
taken ρ = .5 to draw the former picture in contrast with the value ρ = 0 in
the latter picture, while the other parameters are held fixed. Clearly, the
ordinate at the center in the Figure 3.6.1 (a) happens to be 1/2π (≈ .15915)
whereas the ordinate at the center in the Figure 3.6.2 (a) happens to be (1/
2π) (≈ .18378) which is larger. This justifies the preceding claim. One
observes a similar feature when the Figures 3.6.1 (b)-3.6.2 (b)
