Probability definition and properties

The probability is defined as follows [10]:

P(X)=lim

(2.13)

where

P(X) is the occurrence probability of event X.

N is the number of times event X occurs in the n repeated experiments.

Some of the basic probability properties are as follows [7,10]:

• The probability of the occurrence of an event, say event X, is

• The probability of the occurrence and nonoccurrence of an event, say event X, is always:

P(X) + P(X) = 1 (2.15)

where

P(X) is the probability of the occurrence of event X.

P(x) is the probability of the nonoccurrence of event X.

• The probability of an intersection of n independent events is

P(X1X2X3...X„) = P(X1)P(X2)P(X3)...P(X„) (2.16)

where

P(X,) is the occurrence probability of event X„ for i = 1,2,3,..., n.

• The probability of the union of n independent events is

H p(Xl + x2 + — + X„) = 1 - Ujl - P(X,)) (2.17)

i=l

• The probability of the union of n mutually exclusive events is

H

P(X1+X2+ —+ X„) = 2>(X,) (2.18)

i=l

Example 2.3

Assume that an engineering system is composed of two independent subsystems X1 and X2. The failure of either subsystem can result in engineering system failure. The probability of failure of subsystems Xj and X2 is 0.04 and 0.05, respectively.

Calculate the probability of failure of the engineering system.

By substituting the given data values into Equation (2.17), we get

2

P(X,+X2)=l-f[(l-P(X,)) i=l

= P(X1)+P(X2)-P(X1)P(X2)

= 0.04 + 0.05-(0.04)(0.05)

= 0.088

Thus, the probability of failure of the engineering system is 0.088.

Mathematical definitions

This section presents a number of definitions considered useful to perform various types of reliability, maintainability, and safety studies concerned with engineering systems.

Cumulative distribution function

For a continuous random variable, the cumulative distribution function is defined by [9,10].

  • 00 F(')=pWy
  • (2.19)

where

y is a continuous random variable.

f(y) is the probability density function.

F(t) is the cumulative distribution function.

For t = Equation (2.19) becomes

  • 00 F(~)=j/(yMy
  • (2.20)

= i

It means that the total area under the probability density curve is equal to unity.

Generally, in reliability, maintainability, and safety studies of engineering systems, Equation (2.19) is simply written as

i

FW = J/(y)^y (2-21)

o

Example 2.4

Assume that the probability (i.e., failure) density function of an engineering system is

f(t) = ae~M, fort>0,a> (2.22)

where

f(t) is the probability density function (usually, in the area of reliability, it is called the failure density function).

t is a continuous random variable (i.e., time).

a is engineering system failure rate.

Obtain an expression for the engineering system cumulative distribution function by using Equation (2.22).

By substituting Equation (2.22) into Equation (2.21), we obtain

(2.23)

Thus, Equation (2.23) is the expression for the engineering system cumulative distribution function.

Probability density function

For a continuous random variable, the probability density function is expressed by [10]

/(»=«

(2.24)

at

where

f(t) is the density function.

F(t) is the cumulative distribution function.

Example 2.5

Prove with the aid of Equation (2.23) that Equation (2.22) is the probability density function.

By inserting Equation (2.23) into Equation (2.24), we obtain

dt (2-25)

= ae-M

Equations (2.22) and (2.25) are identical.

Expected value

The expected value of a continuous random variable is expressed by [10]

00

E(f) =

  • (2.26)
  • -co

where

E(t) is the expected value (i.e., mean value) of the continuous random variable t.

Example 2.6

Find the expected value (i.e., mean value) of the probability (failure) density function defined by Equation (2.22).

By inserting Equation (2.22) into Equation (2.26), we obtain

E(t) = fte-Mdt

o

(2.27)

Thus, the expected value (i.e., mean value) of the probability (failure) density function defined by Equation (2.22) is given by Equation (2.27).

Laplace transform definition and Laplace transforms of common functions

The Laplace transform (named after a French Mathematician, Pierre-Simon Laplace [1749-1827]) of a function, say f(t), is expressed by [1,11,12]

00

/(s) = p(f)e-df

(2.28)

o

where

s is the Laplace transform variable.

t is a variable.

f(s) is the Laplace transform of function f(t).

Example 2.7

Obtain the Laplace transform of the following function:

(2.29)

where

0 is a constant.

t is a continuous random variable.

Table 2.1 Laplace transforms of some functions

No.

f(t)

f(s)

1

c (a constant)

c

s

2

T

1

s2

3

t", for m = 0,1, 2,3,...

ml sm+^

4

1 s + A.

5

te^

1 (s + X)2

6

tf(t)

df(s) ds

7

9>/l(f) + 02/2(t)

01/l(s) + 02fl(s)

8

df(t) dt

sf(s)-f(O)

By inserting Equation (2.29) into Equation (2.28), we get

co

/(s)=C-e'e-s'df

  • 0
  • (2.30)

Thus, Equation (2.30) is the Laplace transform of Equation (2.29).

Laplace transforms of some commonly occurring functions in engineering system reliability, maintainability, and safety analysis studies are presented in Table 2.1 [11-13].

Final value theorem Laplace transform

If the following limits exist, then the final value theorem may be expressed as

lim/(f) = lim[s/(s)]

(2.31)

t—»00 <—»() *- -1

Example 2.8

Prove, by using the following equation, that the left-hand side of Equation (2.31) is equal to its right side:

f(t= 0(1 + 0(2 „-(«1 +«2 )i

  • (2.32)
  • (oq+a2) (ai+a2)

where

a; and a2 are the constants.

By inserting Equation (2.32) into the left-hand side of Equation (2.31), we obtain

lim +. a2 = 0(1 (2.33)

<->~[cq + a2 (oq+a2) J (oq + a2)

With the aid of Table 2.1, we get the following Laplace transforms of Equation (2.32):

f (s) = , A , ai A -----------r (2.34)

s(cq + a2) (oq+a2) (s + cq + a2)

By substituting Equation (2.34) into the right hand of Equation (2.31), we get

lims

$->0

«1 ! «2__1

s(cq+a2) (oq+a2) (s + cq+a2)

cq (oq + a2)

(2.35)

As the right-hand sides of Equations (2.33) and (2.35) are identical, it proves that the left-hand side of Equation (2.31) is equal to its right side.

 
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