Sin ttulo de diapositiva

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EngineeringComputation
Part 5
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Some Concepts Previous to Probability
RANDOM EXPERIMENT A random experiment or trial
can be thought of as any activity that will
result in one and only one of several
well-defined outcomes, but one does not know in
advance which one will occur.
SAMPLE SPACE The set of all possible outcomes of
a random experiment E, denoted by S(E), is called
the sample space of the random experiment E.
EXAMPLE Suppose that the structural condition of
a concrete structure (e.g., a bridge) can be
classified into one of three categories poor,
fair, or good. An engineer examines one such
structure to assess its condition. This is a
random experiment. Its sample space, S(E)
poor, fair, good, has three elements. If
instead one measures the concrete quality in the
range 0,10, this is the sample space.
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Example of a random experiment
RANDOM EXPERIMENT Rolling two dices
SAMPLE SPACE The set 1,2,3,4,5,6 x
1,2,3,4,5,6
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Random Variable
RANDOM VARIABLE A random variable can be defined
as a real-valued function defined over a sample
space of a random experiment. That is, the
function assigns a real value to every element in
the sample space of a random experiment. The
set of all possible values of a random variable
X, denoted by S(X), is called the support or
range of the random variable X.
EXAMPLE In the previous concrete example, let X
be -1, 0, or 1, depending on whether the
structure is poor, fair, or good, respectively.
Then X is a random variable with support S(X)
-1, 0, 1. The condition of the structure can
also be assessed using a continuous scale, say,
from 0 to 10, to measure the concrete quality,
with 0 indicating the worst possible condition
and 10 indicating the best. Let Y be the assessed
condition of the structure. Then Y is a random
variable with support S(Y ) y 0 y 10
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Random Variable
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Random Variable
NOTATION We consistently use the customary
notation of denoting random variables by
uppercase letters such as X, Y , and Z or X1,X2,
. . . ,Xn, where n is the number of random
variables under consideration. Realizations of
random variables (that is, the actual values they
may take) are denoted by the corresponding
lowercase letters such as x, y, and z or x1, x2,
. . . , xn.
DISCRETE AND CONTINUOUS RANDOM VARIABLES A
random variable is said to be discrete if it can
assume only a finite or countably infinite number
of distinct values. Otherwise, it is said to be
continuous. Thus, a continuous random variable
can take an uncountable set of real values. The
random variable X in the concrete example with
possible values -1. 0, 1 is discrete whereas the
random variable Y, with values in 0,10, is
continuous.
UNIVARIATE AND MULTIVARIATE RANDOM
VARIABLES When we deal with a single random
quantity, we have a univariate random variable.
When we deal with two or more random quantities
simultaneously, we have a multivariate random
variable.
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Probability axioms
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Probability properties
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Induced probability of a random variable
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Induced probability of a random variable
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Induced probability of a random variable
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Conditional probability
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Independence of events
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Total probability and Bayes theorems
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Probability of a random variable

• To specify a random variable we need to know
• its range or support, S(X), which is the set of
  all possible values of the random variable, and
• a tool by which we can obtain the probability
  associated with every subset in its support,
  S(X). These tools are some functions such as the
  probability mass function (pmf), the cumulative
  distribution function (cdf)

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Probability mass function of a discrete random
variable
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Cumulative distribution function of a discrete
random variable
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Moments of a discrete random variable
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Moments of a discrete random variable
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Bernoulli Random Variable
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Binomial random variable
The binomial random variable arises when one
repeats n identical and independent Bernoulli
experiments and observes the number of successes.
EXAMPLESThe number os cars taking left at one
intersection of a series of 100 carsThe number
of broken specimens in a test of a series of 100
specimensThe number of exceedances of a given
flow level in a series of 365 days.The number
of waves higher than 10 m in a series of 1000
waves.
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Binomial random variable
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Binomial random variable
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Binomial random variable
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Binomial random variable
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Geometric or Pascal random variable
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Geometric or Pascal random variable
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Return period
Thus, the return period is 1/p for
exceedancesFor large values it becomes 1/(1-p)
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Negative binomial random variable
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Negative binomial random variable
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Negative binomial random variable
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Hypergeometric random variable
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Hypergeometric random variable
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Poisson random variable
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Poisson random variable
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Poisson random variable
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Poisson random variable
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Multivariate random variable
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Marginal probability mass function
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Conditional probability mass function
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Variance and covariances
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Means, variances and covariances
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Covariance and correlation
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Covariance and correlation
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Multinomial distribution
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Multinomial distribution