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5-3 TWO CONTINUOUS RANDOM VARIABLES 165

fact that these variables are not independent can be determined quickly by noticing that the
range of (X, Y), shown in Fig. 5-8, is not rectangular. Consequently, knowledge of X changes
the interval of values for Y that receives nonzero probability.

EXAMPLE 5-20 Suppose that Example 5-15 is modified so that the joint probability density function of X and Y
6 0.001x 0.002y
is f XY 1x, y2 2 10 e for x 0 and y 0. Show that X and Y are independ-
ent and determine P1X 1000, Y 10002.
The marginal probability density function of X is

f 1x2 2 10 6 0.001x 0.002y dy 0.001 e 0.001x for x 0
e
X
0
The marginal probability density function of y is

6 0.001x 0.002y 0.002y
1 y2 2 10 e dx 0.002 e for y 0
f Y
0
Therefore, f XY (x, y) f X (x) f Y (y) for all x and y and X and Y are independent.
To determine the probability requested, property (4) of Equation 5-21 and the fact that
each random variable has an exponential distribution can be applied.

2
P1X 1000, Y 10002 P1X 10002P1Y 10002 e 1 11 e 2 0.318
Often, based on knowledge of the system under study, random variables are assumed to be in-
dependent. Then, probabilities involving both variables can be determined from the marginal
probability distributions.

EXAMPLE 5-21 Let the random variables X and Y denote the lengths of two dimensions of a machined part, re-
spectively. Assume that X and Y are independent random variables, and further assume that the
2
distribution of X is normal with mean 10.5 millimeters and variance 0.0025 (millimeter) and
that the distribution of Y is normal with mean 3.2 millimeters and variance 0.0036 (millime-
2
ter) . Determine the probability that 10.4 X 10.6 and 3.15 Y 3.25.
Because X and Y are independent,

P110.4 X 10.6, 3.15 Y 3.252 P110.4 X 10.62P13.15 Y 3.252
10.4 10.5 10.6 10.5 3.15 3.2 3.25 3.2
P a Z b P a Z b
0.05 0.05 0.06 0.06
P1 2 Z 22P1 0.833 Z 0.8332 0.566
where Z denotes a standard normal random variable.

EXERCISES FOR SECTION 5-3
5-34. Determine the value of c such that the function (a) P1X 2, Y 32 (b) P1X 2.52
f(x, y) cxy for 0 x 3 and 0 y 3 satisfies the (c) P11 Y 2.52 (d) P1X 1.8, 1 Y 2.52
properties of a joint probability density function. (e) E(X) (f) P1X 0, Y 42
5-35. Continuation of Exercise 5-34. Determine the
following: