09. Probability and Random Variables

8/10/2019 09. Probability and Random Variables

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PROBABILITY AND

RANDOM VARIABLES Ashar Saputra, PhD

List of Refferences

• Johnson, R.A., 2005, Probability and Statistics for

Engineers, Pearson Prenstice Hall, New Jersey, USA

• Walpole, R.E., Myers, R.H., Myers, S.L., Ye, K., 2007,

Probablisty and Statistics for Engineers and Scientists,

Pearson Prenstice Hall, New Jersey, USA

• Madsen, H.O., Krenk, S., Lind, N.C., 1986, Methods of

Structural Safety, Prenstice Hall, New Jersey, USA

• Nowak A.S., Collins K.R., 2000, Reliability of Structures,

McGraw-Hill Int, Singapore

Introduction

What is statistics?• General meaning

• Collection of facts or data – numerical results

• A branch of knowledge

• Scientific meaning – the science of collection,condensation, analysis, interpretation

• Objectives – estimation, prediction, and decision

making

• Risks? Errors might occur (should be accepted)• statistical methods enable to measure or control such

Statistics, as a subject, provides a body of principles and

methodology for designing the process of datacollection, summarizing and interpreting the data, and

drawing conclusions or generalities.

• Why use statistics?

• Universality of variation in all areas

• Estimation, prediction, decision making, etc.(statistics is the science of making decisions in the

face of uncertainty)

Statistics in our everyday life?In our attempt to understand issues of many aspects, numerical

facts and figures need to be reviewed and interpreted. In our day

to day life, learning takes place through an often implicit analysis

of factual information.

Statistics in aid of scientific inquiry?Sciencetific inquiry refers to a systematic process of learning. A

researcher sets the goal of an investigation, collects relevantfactual information (or data), analysis the data, draws

conclusions, and decides further courses of action.

Introduction

• Uncertainties in the building process

• Natural cuses of uncertainty result from the upredictability of

loads such as wind, earthquake, snow, water pressure, or

live load. Other is material used.• Human causes included intended or unintended departures

from design. Examples of uncertainty; approximation,

calculation errors, communicatin problems, omission, etc.

Also inadaquate material, methods of construction, bad

connectins, or change without analysis.

Refreshments

Statistical cases in the field of structural engineering;• Earthquake occurance

• Live load of classrooms per square meter

• Measurements of dimension of beams or columns

• Concrete compressive strength taken from 200

concrete mixers

• Nominal diameter or steel reinforcements

• Weight of truck passing a bridge• Wind speed on certain area

• Properties of organic material

• etc

• Because of these uncertainties, load and resistance (i.e.Load-carrying capacities of structural elements) arerandom variables.

• It is convinient to consider a random parameter (load orresistance) as a function of three factors;

• Physical variation factor. Example include a natural variationof wind pressure, earthquake, live load, and materialproperties.

• Statistical variation factor. This is due to limited sample size.In most situation the natural variation is unknown and it isquantified by examining limited sample data. Therefore, thelarger the sample size, the samller uncertainty described bystatistical variation factor.

• Model variation factor. This is uncertainty due to simplifyingassumption, unknown boundary conditions, and unknowneffects of other variables

Refreshments (LRFD)

PROBABILITY

Probability v.s. Possibility(Oxford Dictionary)

• probability

• · n. (pl. probabilities) the extent to which something is probable. >

a probable or most probable event.

• – PHRASES in all probability most probably.

• probabilistic• · adj. based on or adapted to a theory of probability; involving

chance variation.

• possibility

• · n. (pl. possibilities)• 1 a thing that is possible.

• 2 the state or fact of being possible.

• 3 (possibilities) unspecified qualities of a promising nature.

example

a) 1/16

b) 3/52

c) 3/51

d) 1/4

answer

Example

a) 1/6

b) 1/3c) ½

d) 2/3

e) 5/6

answer

Bayes Theorem

APPLICATION

Hazard Analysis

• An important application area of probability and statistics

is the assessment of natural and man-made risks

• Evaluation of safety of an engineering facility against

extreme environmental actions, such as earthquakes,

strong winds, extreme floods, ocean waves, etc.

• Since environmental loads vary in time, one usually

expresses reliability through the probability that some

undesirable “failure” event (severe structural damage or

collapse, levee breach, dam overtopping, ship hullbuckling, etc.) occurs at least once during a reference

period of time T, for example 50 or 100 years.

Hazard Analysis

In order to quantify risk, one needs to combine two elements:

1. A description of the severity of the environment, in terms of theprobability P[LT > l] with which the maximum environmental load in T

years, LT, exceeds various levels l. Evaluating P[LT > l] is often

referred to as hazard assessment;

2. A description of the resistance of the facility in terms of the

dependence of the probability of system failure Pf on the magnitudel of the environmental load. This function, Pf (l), is often referred to

as the fragility function.

Once quantified, the hazard and fragility functions are combined to

produce the probability of (at least one) failure in T. This is done by

using the Total Probability Theorem, which says that, if {B1,..., Bn} isa set of mutually exclusive and collectively exhaustive events and A is

any other event, then the probability of A can be calculated as

Hazard Analysis

• To use this result for risk assessment, the environmental

load LT is discretized into n distinct levels, say l1, ..., ln and theevents A and Bi in Eq. 1 are taken as

• It is often reasonable to assume that the facility of interestsurvives in T if it does not fail under the most intense load LT

experienced during that period. Under such simplifying

• assumption and with the notation in Eq. 2, the probability offailure in T is obtained from Eq. 1 as

• Eq. 3 shows how the hazard (probabilities P[LT= l

i]) and the

fragility (the probabilities P[failure|LT= l

i]) are combined in the

assessment of risk.

Hazard Analysis

• Example. In a recent study, the seismic hazard in Boston has been assessed as

follows in terms of Modified Mercalli Intensity or MMI (MMI is a discrete scale ofground motion intensity, with integer values from 1 to 12). Over a period of 100

years, the probability that the maximum MMI value equals I is

Hazard Analysis

• In a separate study, the seismic fragility of various types of structures was

assessed by a group of engineers. Some of their results are reproducedbelow in the form of values of the probability of failure for different MMI.

EARTHQUAKE PREDICTION FROM IMPERFECT

PREMONITORY SIGNS

• The main issue that determines the practical usefulness

of these premonitory events is the accuracy with which

predictions can be made. Accuracy can be quantified in

terms of the following probabilities

EARTHQUAKE PREDICTION

• The probabilities in Eq. 1 depend on the strength of the association

between the premonitory event and the occurrence of earthquakes.

Unfortunately, this association is often weak. To make a quantitative

analysis, define the following events:

• A typical daily probability for a major earthquake might be P[E] = 10-5 (hence

P[EC] = 1-10-5), meaning that at a given location large earthquakes might

occur on average every about 300 years. Also, a typical association between

a premonitory event A and large earthquakes might be

• Notice that, for prediction purposes, one looks at the probability of A in the 24

hours that preceed a major earthquake. Applying Bayes’ theorem to this case

• Notice that the probability of a major earthquake, which for a generic day is 10-5,

increases 100-fold, to 10-3, after observation of the premonitory event A. After all,only once every 1000 such warnings, an earthquake would actually occur. Also

consider that issuing false earthquake warnings is very costly and that, following

two or three such erroneous calls, people would lose confidence in the predictions.

• Advocates of earthquake warning have observed that, although single premonitory

signs are seldom useful as a basis for issuing warnings, the use of several such

signs in combination may lead to more accurate predictions. Suppose for example

that two diagnostic events, A and B, are being monitored. For simplicity, assume

that, when taken in isolation, A and B have the same association with E, i.e.

END OF PROBABILITYSESSION

RANDOM VARIABLES

example

09. Probability and Random Variables

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