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160 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

A probability involving only one random variable, say, for example, P 1a X b2,
can be found from the marginal probability distribution of X or from the joint probability

distribution of X and Y. For example, P(a X b) equals P(a X b, Y ).
Therefore,
b b b
P 1a X b2 f 1x, y2 dy dx ° f XY 1x, y2 dy¢ dx f 1x2 dx

XY X

a R x a
a R x
Similarly, E(X) and V(X) can be obtained directly from the joint probability distribution of X
and Y or by first calculating the marginal probability distribution of X. The details, shown in
the following equations, are similar to those used for discrete random variables.

Mean and
Variance from
X
Joint E1X 2 xf 1x2 dx f 1x, y2 dy§ dx
x £
Distribution X XY

R x

xf XY 1x, y2 dx dy (5-17)
R
and

2 2
2
V1X2 x 1x 2 f X 1x2 dx 1x 2 £ f 1x, y2 dy§ dx
X
X XY

R x

1x 2 f XY 1x, y2 dx dy
2
X
R
where R X denotes the set of all points in the range of (X, Y) for which X x and
R Y denotes the set of all points in the range of (X, Y)

EXAMPLE 5-16 For the random variables that denote times in Example 5-15, calculate the probability that Y
exceeds 2000 milliseconds.
(x, y) over the darkly shaded region
This probability is determined as the integral of f XY
in Fig. 5-10. The region is partitioned into two parts and different limits of integration are de-
termined for each part.

2000
6 0.001x 0.002y
P 1Y
20002 ° 6 10 e dy¢ dx
0 2000

6 0.001x 0.002y
° 6 10 e dy¢ dx
2000 x