Page 442 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 13.4 Probability Distribution 429

number, is found from the equation
1 r\(T - "
Prob [-A( t < x(/) <A(t ] / i/,v (13.4-9)

(j 41iT •' -AiJ
— A / r
The following table presents numerical values associated with A = 1,2, and 3.
A Prob[ - Afr < A'(/) < Aijr] P r o b [ |.\ ! > Arr]
68.37r 31.7%
95.4% 4.6%
99.7% 0.3%

The probability of x(r) lying outside ±Arr is the probability of U| exceeding Arr,
which is 1.0 minus the preceding values, or the equation

Prob [U | > Afj] = 10)

Random variables restricted to positive values, such as the absolute value A
of the amplitude, often tend to follow the Rayleigh distribution, which is defined
by the equation
p{A) = — A > 0 (13.4-11)
(T"
The probability density p{A) is zero here for A < 0 and has the shape shown in
Fig. 13.4-6.
The mean and mean square values for the Rayleigh distribution can be found
from the first and second moments to be
_ ^00 A
A = f Ap(A dA = I ~ dA = - i/4
)
■'ll -'ll (r~ V ^
(13.4-12)
A^ = / A^p{A ) dA = I ' dA 2rr"
0 (r
p(A)
0.6
\ Rayleigh distribution
0.5
0.4
0.3
0.2
0.
Figure 13.4-6. Rayleigh dislrihu-
-O tion.