Page 131 - Probability and Statistical Inference

108 3. Multivariate Random Variables

variables, the integral in (3.3.1) will instead represent the total volume under
the density surface z = f(x , x ).
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The marginal distribution of one of the random variables is obtained by
integrating the joint pdf with respect to the remaining variable. The marginal
pdf of X is then formally given by
i

Visualizing the notion of the conditional distribution in a continuous sce-
nario is little tricky. In the discrete case, recall that f (x ) was simply inter-
i
i
preted as P(X = x ) and hence the conditional pmf was equivalent to the
i
i
corresponding conditional probability given by (3.2.4) as long as P(X = x ) >
i
1
0. In the continuous case, however, P(X = x ) = 0 for all x ∈ χ with i = 1,
i
i
i
i
2. So, conceptually how should one proceed to define a conditional pdf?
Let us first derive the conditional df of X given that x ≤ X ≤ x + h where
2
2
2
1
h(> 0) is a small number. Assuming that P(x ≤X ≤x + h) > 0, we have
2 2 2

From the last step in (3.3.3) it is clear that as h ↓ 0, the limiting value of this
conditional probability takes the form of 0/0. Thus, by appealing to the
L’Hôpital’s rule from (1.6.29) we can conclude that

Next, by differentiating the last expression in (3.3.4) with respect to x , one
1
obtains the expression for the conditional pdf of X given that X = x .
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