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4-4 MEAN AND VARIANCE OF A CONTINUOUS RANDOM VARIABLE 105

Determine the probability density function for each of the fol- 4-20.
lowing cumulative distribution functions.
0 x 2
4-18. F1x2 1 e 2x x 0
0.25x 0.5 2 x 1
4-19. F1x2 µ
0.5x 0.25 1 x 1.5
0 x 0 1 1.5 x
0.2x 0 x 4
F1x2 µ 4-21. The gap width is an important property of a magnetic
0.04x 0.64 4 x 9
recording head. In coded units, if the width is a continuous ran-
1 9 x dom variable over the range from 0 x 2 with f(x) 0.5x,
determine the cumulative distribution function of the gap width.

4-4 MEAN AND VARIANCE OF A CONTINUOUS
RANDOM VARIABLE

The mean and variance of a continuous random variable are defined similarly to a discrete
random variable. Integration replaces summation in the definitions. If a probability density
function is viewed as a loading on a beam as in Fig. 4-1, the mean is the balance point.

Definition
Suppose X is a continuous random variable with probability density function f(x).

The mean or expected value of X, denoted as or E(X), is

E1X2 xf 1x2 dx (4-4)

2
The variance of X, denoted as V(X) or , is

2 2 2
2
V1X2 1x
2 f 1x2 dx x f 1x2 dx

The standard deviation of X is 2 2 .

The equivalence of the two formulas for variance can be derived as one, as was done for dis-
crete random variables.

EXAMPLE 4-6 For the copper current measurement in Example 4-1, the mean of X is
20 20
2
E1X2 xf 1x2 dx 0.05x 2 ` 10
0
0
The variance of X is
20 20
2 3
V1X2 1x 102 f 1x2 dx 0.051x 102 3 ` 33.33
0
0