Page 140 - Probability and Statistical Inference

3. Multivariate Random Variables 117

i = 1, 2, 3. With x = (x , x , x ) and , let us denote
1 2 3

Now, one can easily see that

and

Hence, for the function f(x) defined in (3.3.47) we have

and also observe that f(x) is always non-negative. In other words, f(x) is a
pdf of a three dimensional random variable, say, X = (X , X , X ). The joint
3
2
1
pdf of X , X is then given by
1 3

One may directly check by double integration that g(x , x ) is a genuine pdf
1
3
with its support being χ × χ . For any 0 < x < 2, with g(x , x ) from (3.3.49),
3
1
1
3
2
the conditional pdf of X given that X = x , X = x is then given by
2 1 1 3 3