Download Random Variables and Probability Distributions

07-W4959 10/7/08 3:06 PM Page 370
370
Chapter 7
■
Random Variables and Probability Distributions
whereas
P13 x 7 2 p14 2 p15 2 p16 2
However, if x is a continuous random variable, such as task completion time, then
P13 x 7 2 P13 x 7 2
because the area under a density curve and above a single value such as 3 or 7 is 0.
Geometrically, we can think of finding the area above a single point as finding the area
of a rectangle with width 0. The area above an interval of values therefore does not
depend on whether either endpoint is included.
For any two numbers a and b with a b,
P1a x b 2 P1a x b 2 P1a x b 2 P1a x b 2
when x is a continuous random variable.
Probabilities for continuous random variables are often calculated
using cumulative areas. A cumulative area is all of the area under the density curve to the left of a particular value. Figure 7.8 illustrates the cumulative area to the left of .5, which is P(x .5). The probability that x
is in any particular interval, P(a x b), is the difference between two
cumulative areas.
P(x < .5)
.5
F i g u r e 7 . 8 A cumulative area under
a density curve.
The probability that a continuous random variable x lies between a lower limit a and an upper
limit b is
P1a x b 2 1cumulative area to the left of b2 1cumulative area to the left of a 2
P1x b2 P1x a 2
The foregoing property is illustrated in Figure 7.9 for the case of a .25 and
b .75. We will use this result extensively in Section 7.6 when we calculate probabilities using the normal distribution.
For some continuous distributions, cumulative areas can be calculated using methods from the branch of mathematics called integral calculus. However, because we are
not assuming knowledge of calculus, we will rely on tables that have been constructed
for the commonly encountered continuous probability distributions.
F i g u r e 7 . 9 Calculation
of P(a x b) using cumulative areas.
P(.25 < x < .75)
P( x < .75)
P( x < .25)
=
0
.25
.5
.75
1
–
0
.25
.5
.75
1
0
.25
.5
.75
1