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Course: CWR 6536, Spring 2011
School: University of Florida
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Word Count: 809

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of Review Probability Theory CWR 6536 Stochastic Subsurface Hydrololgy Random Variable (r.v.) A variable (x) which takes on values at random, and may be thought of as a function of the outcomes of some random experiment. The r.v. maps sample space of experiment onto the real line The probability with which different values are taken by the r.v. is defined by the cumulative distribution function, F(x), or the...

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of Review Probability Theory CWR 6536 Stochastic Subsurface Hydrololgy Random Variable (r.v.) A variable (x) which takes on values at random, and may be thought of as a function of the outcomes of some random experiment. The r.v. maps sample space of experiment onto the real line The probability with which different values are taken by the r.v. is defined by the cumulative distribution function, F(x), or the probability density function, f(x). Examples Discontinuous r.v. - die tossing experiment Examples Categorical r.v. An observation, s(), that can take on any of a finite number of mutually exclusive, exhaustive states (sk) , e.g. soil type, land use, landscape position An indicator random variable can be defined i ( , sk ) = 1 if s( ) = sk = 0 otherwise The frequency of occurrence of a state f (sk) can be determined as the arithmetic average of n indicator data (i(,sk) )where: 1 n f ( sk ) = i ( , s k ) n =1 The joint frequency of two states sk and vk is 1 n f (s , v ) = i , s k i , vk k k n = 1 ( )( ) Frequency Table for Categorical Data Soil type Sand Silt Loam Clay Frequency 20% 33% 25% 22% Land Use Forest Pasture Meadow Frequency 14% 21% 65% The probability distribution of categorical data is completely described by a frequency table Probability Density Function (pdf) The function f(x) is a pdf for the continuous random variable x, defined over the set of real numbers R, if 1) 2) 3) f ( x) 0 for all x R f ( x)dx = 1 - b P (a < x < b) = f ( x)dx, a i.e. f ( x ) x = P x < x < x + x 0 0 0 [ ] Cumulative Distribution Function (cdf) The cdf of a continuous r.v. x with a pdf f(x) is given by: x F ( x0 ) = P( x x0 ) = f ( x)dx - 0 Therefore Properties of the cdf: F ( x1 ) F ( x 2) for x1 x 2 F ( - ) =0, F ( = , ) 1 dF ( x) f ( x) = dx 0 F ( x1 ) 1 F ( x + =F ( x ) ) b P ( a <x <b) = f ( x ) dx =F(b) - F(a) a Examples: Continuous r.v. uniform distribution, exponential distribution, gaussian distribution log-normal distribution Moments of a Random Variable The pdf (and cdf) summarize all knowledge of the r.v., however we almost never really know this much about actual natural phenomena. Moments of a r.v. provide a more aggregated description of its behavior is which often easier to estimate from field data than pdf or cdf The First Moment The expected value (or first moment, or population mean, or ensemble mean) of a r.v. is defined as the sum of all the values a r.v. may take, each weighted by the probability with which the value is taken x = E ( x) = xf ( x) dx - This quantity is a single valued, deterministic summary of the r.v. Properties of the Expectation Operator If the pdf, f(x) is even (i.e. f(x)=f(-x)), then the expected value is equal to zero The expectation is a linear operator The expected value of a function of the r.v. Higher Order Moments x = E ( x ) = x f ( x)dx - n n n n=1 E[x]=mean (measure of central tendency) n=2 E[x2]= mean square n=3 E[x3]= mean cube E ( x - x) = ( x - x) f ( x)dx n=1 n=2 n=3 n=4 1 = E[( x - x)1] = 0 2 = E[( x - x) 2 ] = 2 3 = E[( x - x)3 ] 4 = E[( x - x) 4 ] [ Central Moments n ] n - variance skewness kurtosis It can be shown that the full (infinite) set of moments completely exhausts the statistical information concerning the r.v., and thus the pdf can be constructed from the full set of moments Joint Probability Distributions The joint cdf for two random variables, x and y, is Fxy ( x0 , y0 ) = P( x x0 , y y0 ) = x0 - - f y0 xy ( x, y )dxdy where Fxy (- , y ) = Fxy ( x,- ) = 0, Fxy (, ) = 1, Marginal cdfs and pdfs cdfs pdfs conditional pdfs Moments of two random variables E [ xy ] = - - xyf xy ( x, y )dxdy Cov ( x, y ) = E ( x - x)( y - y ) = E[ x / y ] = [ ] - - ( x - x)( y - y ) f xy ( x, y )dxdy E [ g ( x, y ) ] = - - xf x / y ( x, y )dx g ( x, y ) f xy ( x, y )dxdy - - Statistical Independence vs Uncorrelation Two random variables are statistically independent if Two random variables are uncorrelated if Two random variables are orthogonal if Statistical Independence vs Uncorrelation If two r.v. are independent then they are uncorrelated (but not vice versa) If x and y are independent random variables then g(x) and h(y) are also independent random variables (this is not generally true if x and y are merely uncorrelated) Correlation measures linear relatedness only An exception is jointly normal r.v.s where uncorrelation is equivalent to independence
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University of Florida - CWR - 6536
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CWR 6536 Stochastic Subsurface Hydrology Optimal Estimation of Hydrologic Parameters using KrigingPurpose of KrigingTo estimate regional distribution of a spatially variable parameter To estimate accuracy of regional distribution Need scattered point me
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CWR 6536 Stochastic Subsurface Hydrology Optimal Estimation of Hydrologic Parameters using KrigingTypes of KrigingSimple kriging is optimal estimation of a random field, e.g. T(x), with a known mean, m(x), and a known covariance PTT(x,x'). Ordinary krig
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Stochastic Modeling Approximate Analytical SolutionsCWR 6536 Stochastic Subsurface HydrologyStochastic model predictions can be obtained in several ways: Exact analytical solutions Monte Carlo techniques Approximate analytical solutions Approximate num
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Approximate Analytical Solutions to the Groundwater Flow ProblemCWR 6536 Stochastic Subsurface Hydrology3-D Steady Saturated Groundwater Flow K K K 0= + + x x y y z z K(x,y,z) random hydraulic conductivity field (x,y,z) random hydraulic head field wa
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University of Florida - EEL - 4930
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Pittsburgh - PSY - 160
Pittsburgh - PSY - 160
Pittsburgh - PSY - 160
Pittsburgh - PSY - 160
Pittsburgh - PSY - 160
Pittsburgh - PSY - 160
Pittsburgh - PSY - 160
Pittsburgh - PSY - 160
Pittsburgh - PSY - 160
Pittsburgh - PSY - 160