Lecture 3 Review Probability

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    CE 5603 SEISMIC HAZARD ASSESSMENT

    LECTURE 3: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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    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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    Total Probability Over a Random Variable

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    Let peak ground acceleration (PGA) be considered as a continuous random

    variable with a median and standard deviation. Also assume that the probability of

    occurence of an event is dependent on the peak ground acceleration expected at

    a site. Then the proper notation for the event can be expressed as:

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    Total Probability Over a Random Variable

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    Total Probability Over a Random Variable

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    Example:

    Two reinforced concrete buildings A and B are located in a seismic region. It isestimated that an impending EQ in the region might be strong (S), moderate (M),

    or weak (W) with probabilities P(S)=0.02, P(M)=0.20, P(W)=0.78. The probabilities

    of failure of each building if these earthquakes occur are 0.20, 0.05 and 0.01

    respectively.

    a) What is the probability of failure of building A if the impending earthquakes

    occur ?

    b) If the building A fails, what is the probability that EQ was of moderate strength ?

    c) Due to similar procedures used in the design and construction of the two

    buildings, it is estimated that if building A fails, the probability that B also fail is 0.5,

    0.15 and 0.02 for 3 types of earthquakes. Determine the probability that both

    buildings will fail in the impending earthquake.

    d) If building A has failed and building B has survived, what is the probability thatEQ was not strong ?

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    Total Probability Over a Random Variable

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    Example:Two reinforced concrete buildings A and B are located in a seismic region. It is

    estimated that an impending EQ in the region might be strong (S), moderate (S),

    or weak (W) with probabilities P(S)=0.02, P(M)=0.20, P(W)=0.78. The probabilities

    of failure of each building if these earthquakes occur are 0.20, 0.05 and 0.01

    respectively.

    a) What is the probability of failure of building A if the impending earthquakes

    occur ?

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    Total Probability Over a Random Variable

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    Example:Two reinforced concrete buildings A and B are located in a seismic region. It is

    estimated that an impending EQ in the region might be strong (S), moderate (S),

    or weak (W) with probabilities P(S)=0.02, P(M)=0.20, P(W)=0.78. The probabilities

    of failure of each building if these earthquakes occur are 0.20, 0.05 and 0.01

    respectively.

    b) If the building A fails, what is the probability that EQ was of moderate strength ?

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    Total Probability Over a Random Variable

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    Example:Two reinforced concrete buildings A and B are located in a seismic region. It is

    estimated that an impending EQ in the region might be strong (S), moderate (S),

    or weak (W) with probabilities P(S)=0.02, P(M)=0.20, P(W)=0.78. The probabilities

    of failure of each building if these earthquakes occur are 0.20, 0.05 and 0.01

    respectively.

    c) Due to similar procedures used in the design and construction of the two

    buildings, it is estimated that if building A fails, the probability that B also fail is 0.5,

    0.15 and 0.02 for 3 types of earthquakes. Determine the probability that both

    buildings will fail in the impending earthquake.

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    Total Probability Over a Random Variable

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    Example:

    d) If building A has failed and building B has survived, what is the probability that

    EQ was not strong ?

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    Poisson Process and Poisson Distribution

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    1. An event can occur at random and at any time or any point in space.

    2. The occurrence(s) of an event in a given time (or space) interval is independent

    of that in any other non-overlapping intervals.

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    Useful Probability Models

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    Useful Probability Models

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    Useful Probability Models

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    Useful Probability Models

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    Widely Used Continuous Distributions

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    Widely Used Continuous Distributions

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    Aleatory Variability and Epistemic

    Uncertainty

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    Aleatory variability is defined as the randomness in the phenomena itself. There's

    no control of uncertainty in the phenomenon and the process is purely random.

    Aleatory variability can not be reduced with more or better observations. One

    solution to decrease aleatory variability inherent in the model may be to completely

    change the model (i.e. switch from empirical observations to a physical model).

    Natural randomness is modeled as a probability density function for the specificproblem being handled.

    Epistemic uncertainty is due to lack of information in how the properties of the

    process changes. With more data and enhanced models, epistemic uncertainty

    can be reduced. Attenuations by proposed by different researchers using the same

    raw sample set, and different best function fits to sample points with small numberof samples will vary the epistemic uncertainty for the problem. Figures 8-10 show

    the seismic source characterizations used by various researchers in nationwide

    seismic hazard mapping studies, forming a typical example for epistemic

    uncertainty definition.

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    Aleatory Variability and Epistemic

    Uncertainty

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    Aleatory Variability and Epistemic

    Uncertainty

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    Aleatory Variability and Epistemic

    Uncertainty

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