Page 182 - Applied statistics and probability for engineers

160 Chapter 5/Joint Probability Distributions

x = Number of Bars of Signal Strength
Marginal
y = Response time Probability
(nearest second) 1 2 3 Distribution of Y
4 0.15 0.1 0.05 0.3
3 0.02 0.1 0.05 0.17
FIGURE 5-6 2 0.02 0.03 0.2 0.25
1 0.01 0.02 0.25 0.28
Marginal probability
distributions of X 0.2 0.25 0.55
and Y from Fig. 5-1. Marginal Probability Distribution of X

For continuous random variables, an analogous approach is used to determine marginal
probability distributions. In the continuous case, an integral replaces the sum.

Marginal
Probability Density If the joint probability density function of random variables X and Y is f XY ( x y), the
,
Function
marginal probability density functions of X and Y are
)
)

5
5
f X ( x) ∫ f XY ( x, y dy and f Y ( y) ∫ f XY ( x, y dx (5-3)
where the irst integral is over all points in the range of ( , ) for which X = x and
Y
X

the second integral is over all points in the range of ( , ) for which Y = y.
Y
X
(
A probability for only one random variable, say, for example, P a, , )
X
b , can be found
from the marginal probability distribution of X or from the integral of the joint probability
distribution of X and Y as
⎤
)
(
(
(
)
x dx 5
⎥
b 5
P a , X , ) ∫ b f X ( ) b ∫ ⎡ ⎢ ∫ ∞ f x, y dy dx 5 b ∞ ∫ ∫ f x, y dydx
∞
a a 2 ⎣ ∞ ⎦ a −∞
Example 5-4 Server Access Time For the random variables that denote times in Example 5-2, calculate the
probability that Y exceeds 2000 milliseconds.
,
This probability is determined as the integral of f XY ( x y) over the darkly shaded region in Fig. 5-7. The region is
partitioned into two parts and different limits of integration are determined for each part.
⎞
⎞
P Y . 2000) 5 2000 ⎛ ⎜ ∫ ∞ 6 310 26 e 20 001 x20 002 y dy dx 1 ∞ ∫ ⎛ ∞ ∫ ⎜ 6 310 e 0 001 2 0 002y dy dx
(
∫
.
.
2
.
.
6 2
x
0
⎟
⎟
0 ⎝ 2000 ⎠ 2000 ⎝ x ⎠

The irst integral is
2000⎛ e 2 0 002y. ∞ ⎞ 63 10 2 6 2000 6 10 2 6 ⎛ 1 e2 2 2 ⎞ ⎞
3
6
2
.
63 10 ∫ ⎜ ⎟ e 2 0 001x. dx 5 e 2 4 ∫ e 2 0 001x dx5 e 2 4 ⎜ ⎟ 50 0475
.
.
.
.
.
0 ⎝ ⎜ 2 0 002 2000⎠ ⎟ 0 0 002 0 0 002 ⎝ 0 001 ⎠
The second integral is
∞ ⎛ e 2 0 002y ∞ ⎞ 63 10 2 6 ∞ 63 10 2 6 ⎛ e 2 6 ⎞
.
2
6
.
.
.
63 10 ∫ ⎜ ⎟ e 2 0 001x dx 5 ∫ e 2 0 003x dx 5 ⎜ ⎟ 5 0 0025
0 002 ⎟
.
.
2000 ⎝ ⎜ 2 . ⎠ 0 0 . 002 2000 0 002 ⎝ 0 003⎠
x