1 engineering computation part 5. 2 some concepts previous to probability random experiment a random...
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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:
1. its range or support, S(X), which is the set of all possible values of the random variable, and
2. 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
PropertiesConcrete example
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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.
EXAMPLES
The number os cars taking left at one intersection of a series of 100 cars
The number of broken specimens in a test of a series of 100 specimens
The 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
Assumptions
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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
Joint probability mass function
Example
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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