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4-5 CONTINUOUS UNIFORM DISTRIBUTION 107

(a) Determine the mean and variance of the coating thickness. (a) Determine the mean and variance of the diameter of the
(b) If the coating costs $0.50 per micrometer of thickness on holes.
each part, what is the average cost of the coating per (b) Determine the probability that a diameter exceeds 5.1 mil-
part? limeters.
4-28. Suppose that contamination particle size (in microm- 4-30. Suppose the probability density function of the length
eters) can be modeled as f 1x2 2x 3 for 1 x. Determine of computer cables is f(x) 0.1 from 1200 to 1210 millime-
the mean of X. ters.
4-29. Integration by parts is required. The probability den- (a) Determine the mean and standard deviation of the cable
sity function for the diameter of a drilled hole in millimeters is length.
10e 101x 52 for x 5 mm. Although the target diameter is 5 (b) If the length specifications are 1195 x 1205
millimeters, vibrations, tool wear, and other nuisances pro- millimeters, what proportion of cables are within specifi-
duce diameters larger than 5 millimeters. cations?

4-5 CONTINUOUS UNIFORM DISTRIBUTION

The simplest continuous distribution is analogous to its discrete counterpart.

Definition
A continuous random variable X with probability density function

f 1x2 1 1b a2, a x b (4-6)

is a continuous uniform random variable.

The probability density function of a continuous uniform random variable is shown in Fig. 4-8.
The mean of the continuous uniform random variable X is

b 2 b
x 0.5x 1a
b2
E1X2 dx `
b a b a a 2
a
The variance of X is

a
b 2 a
b 3 b
b ax a bb ax b 2
2 2 † 1b a2
V1X2 dx
b a 31b a2 a 12
a
These results are summarized as follows.

If X is a continuous uniform random variable over a x b,

1a
b2 1b a2 2
2
E1X 2 and V1X 2 (4-7)
2 12