Lecture 2 Review Probability

7/26/2019 Lecture 2 Review Probability

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CE 5603 SEISMIC HAZARD ASSESSMENT

LECTURE 2:A REVIEW ON PROBABILITY CONCEPTS

By : Prof. Dr. K. nder etin

Middle East Technical University

Civil Engineering Department
http://www.metu.edu.tr/http://www.metu.edu.tr/


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Random Variables

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A random variable is a mapping of the sample space on a real line, such that

every outcome (sample point) in the sample space maps on to a numerical value

on the line representing the corresponding outcome of the random variable. The

mapping need not be one to one, as several outcomes or events in the sample

space may map onto the same point on the line.


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Discrete Random Variables

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An index set may be used to differentiate between various outcomes of a random

variable. For example, X may denote a random variable while X1, X2, X3, etc.

denote its specific outcomes. A random variable is called discrete when its

outcome points on the line are countable, and its called continuous when its

outcome points lie anywhere within one or more intervals on the line.

Example:

Consider the state of a building after an earthquake. The sample space includes

the four outcomes: ND (no damage), LD (light damage), HD (heavy damage), and

C (collapse). No obvious quantitative values are associated with these outcomes.

Hance a random variable may be defined by convention. Consider the random

variable X defined by the following mapping.ND X=0

LD X=1

HD X=2

C X=8


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Probability Mass Function

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Let X be a discrete random variable with possible outcomes, X1, X2, X3, ..... , Xn.

We define the probability mass function (PMF) of X by;

PX (x) = P(X=x)

It is clear that PX (x) =0 for any X that does not coincide with one of the outcomes

X1, X2, X3, ..... , Xn and that PX (xi) = P(X=xi) for any of these outcomes.

Example:

The damage level for a building is presented as a discrete random variable. The

PMF can be plotted as given in Figure 2.

P(ND)= 0.6

P(LD)= 0.3

P(HD)= 0.05P(C)= 0.05

P(X=0)=0.6

P(X=1)=0.3

P(X=2)=0.05

P(X=3)=0.05


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The PMF must obey certain rules;

0Px(x)1 (Probability definition)

This also assures that the PMF to be mutually exclusive and collectively

exhaustive (Figure 3).


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With the PMF given, the probability for any event defined in terms of the random

variable x can be obtained. In particular, the probability that X lies within an

interval (a,b] is given by:

An alternative is to describe cumulative distribution function CDF defined by


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Continuous Random Variables

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A continuous random variable results from the mapping of a continuous sample

space. The random variable may assume any value within one or several intervals

on the line. Since there are infinite points within an interval, the probability that the

random variable will assume any specific value is zero. As also shown in Figure 5,

we define the probability density function (PDF) of a continuous random variable x

as a non-negative function f(x) such that


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It is clear that f(x) is a density quantity since its product with the differential

element dx provides a probability value. Note that probability of occurrence of "x"

within a given distribution is zero; and probability of occurrence of x+dx is

proportional with the shaded area given in Figure 5.

Mutually exclusive and collectively exhaustive property of a continuous randomvariable is expressed as;

Knowing the PDF, we can compute the probability of any event defined in terms

of random variable as such:


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Similar to the case of a discrete random variable, an alternative way to describe

the probability distribution of a continuous random variable is through the

cumulative distribution function.

Hence, knowing the CDF, the PDF can be derived by differentiation


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Example:

Derive PDF and CDF of the distance R from a site to the epicenter of an

earthquake occuring randomly within 100 km of the site. Assume all outcome

points have equal likelihood.


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Example:

Derive the PDF and CDF of the distance R from a site to the epicenter of an

earthquake occuring randomly along a fault. Assume all outcome points along the

fault are equally likely.


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Partial Descriptors of a Random Variable

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A random variable is completely defined by its PMF or PDF. However, often it is

useful to partially characterize a random variable by providing overall features of

its distribution such as the central location, breadth, skewness and other

measures of shape. The mean of x, denoted E(x) or x is defined as the first

moment of its PMF or PDF, i.e;

Another central measure of a random variable is the median. Denoted X0.5, the

median is such that 50% of outcomes lie below it and 50% above it. For a given

random variable, the median is obtained by solving the equation


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A third central measure is the mode. Denoted the mode is the outcome that

has the highest probability or probability density. It is obtained by maximizing p(x)

or f(x).


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Lecture 2 Review Probability