1. -Exponential Distribution -Weibull Distribution
Systems Engineering Program
Department of Engineering Management, Information and Systems
EMIS 7370/5370 STAT 5340 :
PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS
Special Continuous Probability Distributions
-Exponential Distribution
-Weibull Distribution
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
Stracener_EMIS 7370/STAT 5340_Sum 08_

2. Exponential Distribution

3. The Exponential Model - Definition
A random variable X is said to have the Exponential
Distribution with parameters , where  > 0, if the
probability density function of X is:
, for  0
, elsewhere

4. Properties of the Exponential Model
Probability Distribution Function
for < 0
for  0
*Note: the Exponential Distribution is said to be
without memory, i.e.
P(X > x1 + x2 | X > x1) = P(X > x2)

5. Properties of the Exponential Model
Mean or Expected Value
Standard Deviation
Properties of the Exponential Model

6. Exponential Model - Example
Suppose the response time X at a certain on-line computer terminal (the elapsed time between the end of a user’s inquiry and the beginning of the system’s response to that inquiry) has an exponential distribution with expected response time equal to 5 sec. The E(X) = 5=θ, so λ = 0.2.
(a) What is the probability that the response time is at most 10 seconds?
(b) What is the probability that the response time is between 5 and 10 seconds?
(c) What is the value of x for which the probability of exceeding that value is 1%?

7. Exponential Model - Example
The probability that the response time is at most 10 sec is:
The probability that the response time is between 5 and 10 sec is:

8. Exponential Model - Example
The value of x for which the probability of exceeding x is 1%:

9. Weibull Distribution

10. The Weibull Probability Distribution Function
Definition - A random variable X is said to have the Weibull Probability Distribution with parameters  and , where  > 0 and  > 0, if the probability density function of is:
, for  0
, elsewhere
Where,  is the Shape Parameter,  is the Scale
Parameter. Note: If  = 1, the Weibull reduces to
the Exponential Distribution.

11. The Weibull Probability Distribution Function
Probability Density Function
f(t)
1.8
β=5.0
1.6
β=0.5
1.4
β=3.44
1.2
β=1.0
1.0
β=2.5
0.8
0.6
0.4
0.2 t
t is in multiples of 

12. The Weibull Probability Distribution Function
for x  0
b = 5
b = 3
b = 1
F(x)
b = 0.5

13. Weibull Probability Paper (WPP)
Derived from double logarithmic transformation of
the Weibull Distribution Function.
Of the form
where
Any straight line on Weibull Probability paper is a Weibull
Probability Distribution Function with slope,  and intercept,
- ln , where the ordinate is ln{ln(1/[1-F(t)])} the abscissa is
ln t.
Weibull Probability Paper (WPP)

14. Weibull Probability Paper (WPP)
Weibull Probability Paper links

15. Use of Weibull Probability Paper
99.0
95.0
90.0
80.0
70.0
50.0
40.0
30.0
20.0
10.0
5.0
4.0
3.0
2.0
1.0
0.5
Cumulative probability in percent
F(x) in %
1.8 in. = b
1 in. q x

16. Properties of the Weibull Distribution
100pth Percentile
and, in particular
Mean or Expected Value
Note: See the Gamma Function Table to obtain values of (a)
Properties of the Weibull Distribution

17. Properties of the Weibull Distribution
Standard Deviation of X
where

18. The Gamma Function 
Values of the
Gamma Function

19. Properties of the Weibull Distribution
Mode - The value of x for which the probability
density function is masimum
i.e.,
xmode
f(x) x
Max f(x)=f(xmode)

20. Weibull Distribution - Example
Let X = the ultimate tensile strength (ksi) at -200
degrees F of a type of steel that exhibits ‘cold
brittleness’ at low temperatures. Suppose X has a
Weibull distribution with parameters  = 20,
and  = 100. Find:
(a) P( X  105)
(b) P(98  X  102)
(c) the value of x such that P( X  x) = 0.10

21. Weibull Distribution - Example Solution
(a) P( X  105) = F(105; 20, 100)
(b) P(98  X  102)
= F(102; 20, 100) - F(98; 20, 100)

22. Weibull Distribution - Example Solution
(c) P( X  x) = 0.10
P( X  x)
Then

23. Weibull Distribution - Example
The random variable X can modeled by a Weibull distribution with  = ½ and  = The spec time limit is set at x = What is the proportion of items not meeting spec?

24. Weibull Distribution - Example
The fraction of items not meeting spec is
That is, all but about 13.53% of the items will not meet spec.

25. An Application of Probability to Reliability Modeling and Analysis

26. Reliability Definitions and Concepts
Figures of merit
Failure densities and distributions
The reliability function
Failure rates
The reliability functions in terms of the failure rate
Mean time to failure (MTTF) and mean time between failures (MTBF)

27. What is Reliability? Reliability is performance over time, probability
Reliability is defined as the probability that an item will perform its intended function for a specified interval under stated conditions. In the simplest sense, reliability means how long an item (such as a machine) will perform its intended function without a breakdown.
Reliability: the capability to operate as intended, whenever used, for as long as needed.
Reliability is performance over time, probability
that something will work when you want it to.

28. Reliability Figures of Merit
Basic or Logistic Reliability
MTBF - Mean Time Between Failures
measure of product support requirements
Mission Reliability
Ps or R(t) - Probability of mission success
measure of product effectiveness

30. “If I had only one day left to live,
Reliability Humor: Statistics
“If I had only one day left to live,
I would live it in my statistics class --
it would seem so much longer.”
From: Statistics A Fresh Approach
Donald H. Sanders
McGraw Hill, 4th Edition, 1990

31. The Reliability Function
The Reliability of an item is the probability that the item will
survive time t, given that it had not failed at time zero, when
used within specified conditions, i.e.,

32. Reliability
Relationship between failure density and reliability

33. Relationship Between h(t), f(t), F(t) and R(t)
Remark: The failure rate h(t) is a measure of proneness to
failure as a function of age, t.

34. The Reliability Function
The reliability of an item at time t may be expressed in terms
of its failure rate at time t as follows:
where h(y) is the failure rate

35. Mean Time to Failure and Mean Time Between Failures
Mean Time to Failure (or Between Failures) MTTF (or MTBF)
is the expected Time to Failure (or Between Failures)
Remarks:
MTBF provides a reliability figure of merit for expected failure
free operation
MTBF provides the basis for estimating the number of failures in
a given period of time
Even though an item may be discarded after failure and its mean
life characterized by MTTF, it may be meaningful to
characterize the system reliability in terms of MTBF if the
system is restored after item failure.

36. Relationship Between MTTF and Failure Density
If T is the random time to failure of an item, the
mean time to failure, MTTF, of the item is
where f is the probability density function of time
to failure, iff this integral exists (as an improper
integral).

37. Relationship Between MTTF and Reliability

38. Reliability “Bathtub Curve”

39. Reliability Humor

40. The Exponential Model: (Weibull Model with β = 1)
Definition
A random variable T is said to have the Exponential
Distribution with parameters , where  > 0, if the
failure density of T is:
, for t  0
, elsewhere

41. Probability Distribution Function
Weibull W(b, q)
, for t  0
Where F(t) is the population proportion failing in time t
Exponential E(q) = W(1, q)

42. The Exponential Model Remarks
The Exponential Model is most often used in
Reliability applications, partly because of mathematical
convenience due to a constant failure rate.
The Exponential Model is often referred to as the
Constant Failure Rate Model.
The Exponential Model is used during the ‘Useful Life’
period of an item’s life, i.e., after the ‘Infant Mortality’
period before Wearout begins.
The Exponential Model is most often associated with
electronic equipment.

43. Reliability Function Probability Distribution Function Weibull
Exponential

44. The Weibull Model - Distributions
Reliability Functions
R(t) t
t is in multiples of 
β=5.0
β=1.0
β=0.5
1.0
0.8
0.6
0.4
0.2

45. Mean Time Between Failure - MTBF
Weibull
Exponential

46. The Gamma Function 
Values of the
Gamma Function

47. Percentiles, tp
Weibull
and, in particular
Exponential

48. Failure Rates - Weibull
a decreasing function of t if  < 1 Notice that h(t) is a constant if  = 1
an increasing function of t if  > 1
Cumulative Failure Rate
The Instantaneous and Cumulative Failure Rates, h(t)
and H(t), are straight lines on log-log paper.

49. Failure Rates - Exponential
Note:
Only for the Exponential Distribution
Cumulative Failure

50. The Weibull Model - Distributions
Failure Rates
h(t) t
t is in multiples of  h(t) is in multiples of 1/  3 2 1
β=5
β=1
β=0.5

51. The Binomial Model - Example Application 1
Problem -
Four Engine Aircraft
Engine Unreliability Q(t) = p = 0.1
Mission success: At least two engines survive
Find RS(t)

52. The Binomial Model - Example Application 1
Solution -
X = number of engines failing in time t
RS(t) = P(x  2) = b(0) + b(1) + b(2)
= =

53. Series Reliability Configuration
Simplest and most common structure in reliability analysis.
Functional operation of the system depends on the successful operation of all system components Note: The electrical or mechanical configuration may differ from the reliability configuration
Reliability Block Diagram
Series configuration with n elements: E1, E2, ..., En
System Failure occurs upon the first element failure E1 E2 En

54. Series Reliability Configuration with Exponential Distribution
Reliability Block Diagram
Element Time to Failure Distribution
with failure rate , for i=1, 2,…, n
System reliability
where E1 E2 En
is the system failure rate
System mean time to failure

55. Series Reliability Configuration
Reliability Block Diagram
Identical and independent Elements
Exponential Distributions
Element Time to Failure Distribution
with failure rate
System reliability E1 E2 En

56. Series Reliability Configuration
System mean time to failure
Note that q/n is the expected time to the first failure, E(T1), when n identical items are put into service

57. Parallel Reliability Configuration – Basic Concepts
Definition - a system is said to have parallel reliability configuration if the system function can be performed by any one of two or more paths
Reliability block diagram - for a parallel reliability configuration consisting of n elements, E1, E2, ... En E1 E2 En

58. Parallel Reliability Configuration
Redundant reliability configuration - sometimes called a redundant reliability configuration. Other times, the term ‘redundant’ is used only when the system is deliberately changed to provide additional paths, in order to improve the system reliability
Basic assumptions
All elements are continuously energized starting at time t = 0
All elements are ‘up’ at time t = 0
The operation during time t of each element can be described
as either a success or a failure, i.e. Degraded operation or
performance is not considered

59. Parallel Reliability Configuration
System success - a system having a parallel reliability configuration operates successfully for a period of time t if at least one of the parallel elements operates for time t without failure. Notice that element failure does not necessarily mean system failure.

60. Parallel Reliability Configuration Block Diagram
System reliability - for a system consisting of n elements, E1, E2, ... En E1 E2 En
if the n elements operate independently of each other and where Ri(t) is the reliability of element i, for i=1,2,…,n

61. System Reliability Model - Parallel Configuration
Product rule for unreliabilities
Mean Time Between System Failures

62. Parallel Reliability Configurations
p=R(t)

63. Parallel Reliability Configuration with Exponential Distribution
Element time to failure is exponential with failure rate 
Reliability block diagram:
Element Time to Failure Distribution
with failure rate for I=1,2. E1 E2
System reliability
System failure rate

64. Parallel Reliability Configuration with Exponential Distribution
System Mean Time Between Failures:
MTBFS = 1.5 

65. Example A system consists of five components connected as shown.
Find the system reliability, failure rate, MTBF, and MTBM if Ti~E(λ) for i=1,2,3,4,5 E2 E1 E3 E4 E5

66. Solution
This problem can be approached in several different ways. Here is one approach:
There are 3 success paths, namely,
Success Path Event
E1E2 A
E1E3 B
E4E5 C
Then Rs(t)=Ps=
=P(A)+P(B)+P(C)-P(AB)-P(AC)-P(BC)+P(ABC)
=P(A)+P(B)+P(C)-P(A)P(B)-P(A)P(C)-P(B)P(C)+
P(A)P(B)P(C)
=P1P2+P1P3+P4P5-P1P2P3-P1P2P4P5 -P1P3P4P5+P1P2P3P4P5
assuming independence and where Pi=P(Ei) for i=1, 2, 3, 4, 5

67. Since Pi=e-λt for i=1,2,3,4,5
Rs(t)
hs(t)

68. MTBFs