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3. Multivariate Random Variables 133

are visually compared.

How can one show directly that f(x , x ) from (3.6.1) is
2
1
indeed a genuine pdf? The derivation follows shortly.
The function f(x , x ) is always positive. So, we merely need to verify that
2
1
2
the double integral of f(x , x ) over the whole space ℜ is unity. With u , u 2
1
1
2
from (3.6.2), let us then rewrite

Hence, with c defined in (3.6.2) we obtain

Now, for all fixed x ∈ ℜ, let us denote
2

so that with we obtain

Next, look at the expression of ag(x , x ) obtained from (3.6.5) and note that
1
2
for all fixed x , it resembles the pdf of a univariate normal variable with mean
2
µ + ρσ (x – µ )/σ and variance at the point x ∈ ℜ. Hence, we
1
1
1
2
2
2
must have