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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Probability Mass Function
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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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Probability Mass Function
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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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Probability Mass Function
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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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Continuous Random Variables
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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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Continuous Random Variables
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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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Continuous Random Variables
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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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Continuous Random Variables
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Continuous Random Variables
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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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Partial Descriptors of a Random Variable
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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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Partial Descriptors of a Random Variable
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Partial Descriptors of a Random Variable
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Partial Descriptors of a Random Variable
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