Advanced Functions of Temporal GIS
Advanced Functions of Temporal GIS ENVR 468
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This 13 page Class Notes was uploaded by Itzel Hilll on Sunday October 25, 2015. The Class Notes belongs to ENVR 468 at University of North Carolina - Chapel Hill taught by Marc Serre in Fall. Since its upload, it has received 77 views. For similar materials see /class/228866/envr-468-university-of-north-carolina-chapel-hill in Environment at University of North Carolina - Chapel Hill.
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Date Created: 10/25/15
Introduction to Random Variables 1 Random variables 11 A random variable and its realization x is a random variable that takes different possible values 1 is a specific value or realization of x Example x is the concentration of particulate matter concentration in Chapel Hill on July 6 1988 Z is the concentration measured at a monitoring station in Chapel Hill on July 6 1988 by collecting particulate matter dust in a filter and sending the filter to a lab for analysis 12 The cdf of a random variable Let PA be the probability that the event A occurs Note We always have PAe 01 Then the cumulative density function cdf of a random variable x is defined as FM PxS 25 Important properties F xoo 0 Why F xoo 1 Why FxOg is always increasing Proof bgta PxltbPxltaPa xltb Z Pxlta Fxb2Fxa 13 The pdf of a random variable De nition The probability density function pdf of a random variable may be defined as the derivative of its cdf Fx 2 Mr a Z Important properties 0 Since Fx is always increasing then f 9020 V1 EWMXWwaKJWWEWM6M KWHKEW In other words the area under the ag curve between a and b is the probability of the event altx5 3 Hence the pdf is really a density of probability or a probability density function 0 The normalization constraint We must always have f dz fx1 1 mw WAwwmwmfnm mkl 14 The expected value of a random variable The expected value ofgx is Egx dz gfx 5 Examples The mean is the expected value of x mxEx I dz 2 122 The variance is the expected value of x mx2 varxEiltx mx2 I 6172505 mx 2 122 The expected value of x mx4 is Eltx mxgt4 I dz 1 mx 4 122 15 Exercises k if I 6 ab 0 otherwise ow Let fxUt 1 Use the normalization constraint to nd k so that j z is a pdf lfdefAz 1 k1ba 2 Assume that f is the pdf of a random value x What is the expected value of x Eixi 1 aim22 aw2 3 Write the formulae for the variance of x abj2 l varxExmc2f dZCZ mx 2 fx 1 5 61725 2 b a 2 Bivariate distributions x and x are two random variables that may be dependent on one another Example x is the lead concentration in the drinking water at the tap of a house x is the lead concentration in the blood of the person living in that house 21 The bivariate cdf me ZQPszAND x SH Pxsx x SH 22 The bivariate pdf 6 a fxx Z1Z9a Fxx397 z 61 23 Normalization constraint fode iidz39fxxlzw 1 24 The marginal and conditional pdf The marginal pdf offxxx y with respect to Z is c1f0wdz39fo I 1quot The marginal pdf offxxx y with respect to 1 is c19fo dz fxx 139 The conditional pdf x g x 9 ofx given that x is fxxvat39 fxx ZZ39 fxvz39 fwdzfxxz 139 mm x 29 25 The expected value Egxx 9F dz I dz39gCZ z39fxzi 2539 Examples The expected value ofx is mxEx I dz I dz39zfxxKZ 1quot The expected value ofx is mxiEx fO dz fowdz39z39fxint 1quot The variance oix is Eltx mxgt2 I dz 11mg may fxa a 239 The variance ofx is me mxi2 I dz I claw max faxiz 239 The covariance between x and x is Ewe mace mxigt If dz If dz39Ut mx xz ma gt12va 25 z39 26 Exercises k if zeabandl1Eab O Ow Let nyan 1 Use the nonnalization constraint to nd k so that cy Z w is a pdf 0 dz 0 dy fxyog w 1 k1ba2 2 Assume that fxy Z w is the pdf of x and y What is the expected value of x Ex fode fidw cywm aw2 3 Calculate the covariance between x and y covltxyEltx mxgtlty my1 de ffdvz mxxzx my ltb agt2 o Note that varxVary ba212 4 Write the formulae for E 2 EM l dz fwdvwfwww ab2 5 Find the marginal pdf ag of the random variable x b l l 00 fadw Zz if aszltb fxzfwdw 6yzw b a 5 01 0 otherwise 6 Calculate Ex using fx Ex ldzzf z ab2 same as in 2 7 Find the marginal pdf fyl of the random variable y b 1 1 d I Zb a2 b 0 otherwise a if aSyltb my If dz fxyoaw 8 Find the conditional pdf of x given that y 1 fx 1910 if aszltb and aswltb fxiyzyw y b a fyOO 0 otherwise 9 Find the probability that x ltab2 given that yb ab 2 ab 2 Pxltltabgt2yb I dzfxiyUtlybgt f dz ltb agt12 ll Exercises on Conditional probability using discrete variables Leta be a random variable taking values a a2 an Let b be a random variable taking values 31 b2 bm Let Pal represent the probability that aai Let Pb represent the probability that bbj Let Pal 31 represent the probabilities that aai AND bbj Then the probability Pal 31 that aai GIVEN that bbj is Plaz 5 Paz 5 Pbj Example 1 Consider the case where 01 takes values a or 012 and 3 takes values I or 32 In an eXperiment we record over 1000 trials the values for a and b and we obtain the following distribution b1 2 a 100 400 012 100 400 Number of trials Pa1 Pa2 Pb1 Pb2 Fla b1 Pa13 32 Pa1 l b1 Pa1 32 Example 2 Redo the example with the following distribution b1 32 a 400 100 012 100 400 Number of trials Pa1 Pa2 Pb1 Pb2 Fla b1 Pa13 32 Pa1 l b1 Pa1 32 Note that in this example the conditional probability did change while that was not the case in Example 1 Why The change in probability can be though of as knowledge updating Pa1 is the prior probability while Pa1 1 is the updated probability when we know that bb1