Page 371 - Mechanical Engineers' Handbook (Volume 2)

362 Mathematical Models of Dynamic Physical Systems

the probability of that value. The probability distribution function F(x) of a continuous
random variable x is defined as the probability that x assumes a value no greater than x, that
is,
F(x) Pr(X x) x

ƒ(x) dx
The probability density function ƒ(x) is defined as the derivative of F(x).
The mean or expected value of a probability distribution is defined as
E(X)

xƒ(x) dx X
The mean is the first moment of the distribution. The nth moment of the distribution is
defined as
E(X ) n
n
x ƒ(x) dx
The mean square of the difference between the random variable and its mean is the variance
or second central moment of the distribution,

(X) E(X X) 2 2 2
2
2
(x X)ƒ(x) dx E(X ) [E(X)]
The square root of the variance is the standard deviation of the distribution:

2

(X) E(X ) [E(X)] 2
The mean of the distribution therefore is a measure of the average magnitude of the random
variable, while the variance and standard deviation are measures of the variability or dis-
persion of this magnitude.
The concepts of probability can be extended to more than one random variable. The
joint distribution function of two random variables x and y is defined as
F(x,y) Pr(X x and Y y) y
x
ƒ(x,y) dy dx
where ƒ(x,y) is the joint distribution. The ijth moment of the joint distribution is
E(XY ) j

ij
i
x y ƒ(x,y) dy dx
The covariance of x and y is defined to be
E[(X X)(Y Y)]

and the normalized covariance or correlation coefficient as
E[(X X)(Y Y)]

2
2

(X)
(Y)
Although many distribution functions have proven useful in control engineering, far and
away the most useful is the Gaussian or normal distribution