# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

9 Pages

### lecture-35

Course: STAT 36-754, Spring 2006
School: Michigan
Rating:

Word Count: 2330

#### Document Preview

35 Large Chapter Deviations for Stochastic Dierential Equations This last chapter revisits large deviations for stochastic dierential equations in the small-noise limit, rst raised in Chapter 22. Section 35.1 establishes the LDP for the Wiener process (Schilders Theorem). Section 35.2 proves the LDP for stochastic dierential equations where the driving noise is independent of the state of the process. Section...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Michigan >> Michigan >> STAT 36-754

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
35 Large Chapter Deviations for Stochastic Dierential Equations This last chapter revisits large deviations for stochastic dierential equations in the small-noise limit, rst raised in Chapter 22. Section 35.1 establishes the LDP for the Wiener process (Schilders Theorem). Section 35.2 proves the LDP for stochastic dierential equations where the driving noise is independent of the state of the process. Section 35.3 states the corresponding result for SDEs when the noise is state-dependent, and gestures in the direction of the proof. In Chapter 22, we looked at how the diusions X which solve the SDE dX = a(X )dt + dW, X (0) = x0 (35.1) converge on the tra jectory x0 (t) solving the ODE dx = a(x(t)), x(0) = x0 dt (35.2) in the small noise limit, 0. Specically, Theorem 256 gave a (fairly crude) upper bound on the probability of deviations: lim 0 2 log P sup (t) > 0tT 2 e2Ka T (35.3) where Ka depends on the Lipschitz coecient of the drift function a. The theory of large deviations for stochastic dierential equations, known as FreidlinWentzell theory for its original developers, shows that, using the metric implicit 238 239 CHAPTER 35. FREIDLIN-WENTZELL THEORY in the left-hand side of Eq. 35.3, the family of processes X obey a large deviations principle with rate 2 , and a good rate function. (The full Freidlin-Wentzell theory actually goes somewhat further than just SDEs, to consider small-noise perturbations of dynamical systems of many sorts, perturbations by Markov processes (rather than just white noise), etc. Time does not allow us to consider the full theory (Freidlin and Wentzell, 1998), or its many applications to nonparametric estimation (Ibragimov and Hasminskii, 1979/1981), systems analysis and signal processing (Kushner, 1984), statistical mechanics (Olivieri and Vares, 2005), etc.) As in Chapter 31, the strategy is to rst prove a large deviations principle for a comparatively simple case, and then transfer it to more subtle processes which can be represented as appropriate functionals of the basic case. Here, the basic case is the Wiener process W (t), with t restricted to the unit interval [0, 1]. 35.1 Large Deviations of the Wiener Pro cess We start with a standard d-dimensional Wiener process W , and consider its dilation by a factor , X (t) = W (t). There are a number of ways of establishing that X obeys a large deviation principle as 0. One approach (see Dembo and Zeitouni (1998, ch. 5) starts with establishing an LDP for continuous-time random walks, ultimately based on the Grtner-Ellis Theorem, and then showa ing that the convergence of such processes to the Wiener process (the Functional Central Limit Theorem, Theorem 174 of Chapter 16) is suciently fast that the LDP carries over. However, this approach involves a number of surprisingly tricky topological issues, so I will avoid it, in favor of a more probabilistic path, marked out by Freidlin and Wentzell (Freidlin and Wentzell, 1998, sec. 3.2). Until further notice, w will denote the supremum norm in the space of continuous curves over the unit interval, C([0, 1], Rd ). Denition 459 (Cameron-Martin Spaces) The Cameron-Martin space HT consists of al l continuous sample paths x C([0, T ], Rd ) where x(0) = 0, x is absolutely continuous, and its Radon-Nikodym derivative x is square-integrable. Lemma 460 Cameron-Martin spaces are Hilbert spaces, with norm x T 2 |x(t)| dt. 0 CM = Proof: An exercise (35.1) in verifying that the axioms of a Hilbert space are satised. Denition 461 The eective Wiener action of an continuous function x C([0, t], Rd ) is 1 2 x CM (35.4) JT (x) 2 240 CHAPTER 35. FREIDLIN-WENTZELL THEORY if x HT , and otherwise. In particular, J1 (x) 1 2 1 0 2 |x(t)| dt (35.5) For every j > 0, let LT (j ) = {x : JT (x) j }. Prop osition 462 Fix a function f H1 , and let Y = X f . Then L (Y ) = is absolutely continuous with respect to L (X ) = , and the Radon-Nikodym derivative is 1 d ( w) = exp d 1 0 w(t) dW 1 22 1 0 2 |w(t)| dt (35.6) Proof: This is a special case of Girsanovs Theorem. See Corollary 18.25 on p. 365 of Kallenberg, or, more transparently perhaps, the relevant parts of Liptser and Shiryaev (2001, vol. I). Lemma 463 For any , , K > 0, there exists an P( X x ) e 0 > 0 such that, if J1 (x)+ 2 < 0, (35.7) provided x(0) = 0 and J1 (x) < K . Proof: Using Proposition 462, P( X x ) = P( Y 0 = w = e < ) d ( w)d ( w) d J1 (x) 2 e w < 1 R1 0 xdW (35.8) (35.9) d (w) (35.10) From Lemma 254 in Chapter 22, we can see that P ( W < ) 1 as 0. So, if is suciently small, P ( W < ) 3/4. Now, applying Chebyshevs inequality to the integrand, 11 22 P x dW J1 (x) (35.11) 0 1 P 2 0 1 0 E = 1 x dW x dW 8 2 J1 (x) 1 0 22 J1 (x) (35.12) 2 (35.13) 2 |x| dt 1 = 8J1 (x) 4 (35.14) 241 CHAPTER 35. FREIDLIN-WENTZELL THEORY using the It isometry (Corollary 196). Thus o P e e w < 1 1 R1 0 R1 xdW 0 xdW P( X x e 22 3 4 (35.15) 1 22 J1 (x) e 2 1 J1 (x) 22 J1 (x) 2 e 2 d (w) > J1 (x) ) > (35.16) (35.17) where the second term in the exponent can be made less than any desired by taking small enough. Lemma 464 For every j > 0, > 0, let U (j, ) be the open neighborhood of L1 (j ), i.e., al l the trajectories coming within of a trajectory whose action is less than or equal to j . Then for any > 0, there is an 0 > 0 such that, if < 0 and s P (X U (j, )) e 2 (35.18) Proof: Basically, approximating the Wiener process by a continuous piecewiselinear function, and showing that the approximation is suciently ne-grained. Chose a natural number n, and let Yn, (t) be the piecewise linear random function which coincides with X at times 0, 1/n, 2/n, . . . 1, i.e., Yn, (t) = X ([tn]/n) + t [tn] n X ([tn + 1]/n) (35.19) We will see that, for large enough n, this is exponentially close to X . First, though, lets bound the probability in Eq. 35.18. P (X U (j, )) = P X U (j, ), X Yn, +P X U (j, ), X Yn, P X U (j, ), X Yn, P (J1 (Yn, ) > j ) + P < < +P X Yn, (35.20) X Yn, (35.21) (35.22) J1 (Yn, ) can be gotten at from the increments of the Wiener process: J1 (Yn, ) =n = 2 2 n 2 i=1 |W (i/n) W ((i 1)/n)| 2 dn 2 i i=1 (35.23) (35.24) CHAPTER 35. FREIDLIN-WENTZELL THEORY 242 where the i have the 2 distribution with one degree of freedom. Using our results on such distributions and their sums in Ch. 22, it is not hard to show that, for suciently small , P (J1 (Yn, ) > j ) 1 j e2 2 (35.25) To estimate the probability that the distance between X and Yn, reaches or exceeds , start with the independent-increments property of X , and the fact that the two processes coincide when t = i/n. P X Yn, n max P (i1)/nti/n i=1 = nP = nP nP |X (t) Yn, (t)| max |X (t) Yn, (t)| (35.27) max | W (t) n W (1/n)| (35.28) 0t1/n 0t1/n max | W (t)| 0t1/n 4dnP W1 (1/n) 2 (35.29) 2d (35.30) n 2 2d 4dn e 8d2 2 n (35.31) again 2 freely using our calculations from Ch. 22. If n > 4d2 j / 2 , then P j 12 , 2e (35.26) and we have overall P (X U (j, )) e j 2 X Yn, (35.32) as required. Prop osition 465 The Cameron-Martin norm has compact level sets. Proof: See Kallenberg, Lemma 27.7, p. 543. Theorem 466 (Schilders Theorem) If W is a d-dimensional Wiener process on the unit interval, then X = W obeys an LDP on C([0, 1], Rd ), with rate 2 and good rate function J1 (x), the eective Wiener action over [0, 1]. Proof: It is easy to show that Lemma 463 implies the large deviation lower bound for open sets. (Exercise 35.2.) The tricky part is the upper bound. Pick any closed set C and any > 0. Let s = J1 (C ) . By Lemma 465, the set K = L1 (s) = {x : J1 (x) s} is compact. By construction, C K = . So 243 CHAPTER 35. FREIDLIN-WENTZELL THEORY = inf xC,yK x y use Lemma 464 > 0. Let U be the closed -neighborhood of K . Then P (X C ) P (X U ) s 2 e e lim sup 2 0 (35.34) (35.35) J1 (C ) 2 (35.36) log P (X C ) J1 (C ) 2 log P (X C ) J1 (C ) 2 (35.37) (35.38) log P (X C ) 2 J1 (C )2 2 (35.33) 2 Since was arbitrary, this completes the proof. Remark: The trick used here, about establishing results like Lemmas 464 and 463, and then using compact level sets to prove large deviations, works more generally. See Theorem 3.3 in Freidlin and Wentzell (1998, sec. 3.3). Corollary 467 Schilders theorem remains true for Wiener processes on [0, T ], for al l T > 0, with rate function JT , the eective Wiener action on [0, T ]. Proof: If W is a Wiener process on [0, 1], then, for every T , S (W ) = T W (t/T ) is a Wiener process on [0, T ]. (Show this!) Since the mapping S is continuous from C([0, 1], Rd ) to C([0, T ], Rd ), by the Contraction Principle (Theorem 410) the family S (W ) obey an LDP with rate 2 and good rate function JT (x) = J1 (S 1 (x)). (Notice that S is invertible, so S 1 (x) is a function, not a set of functions.) Since x H1 i y = S (x) HT , its easy to check that for such, y (t) = T 1/2 x(t/T ), meaning that y CM T = 0 T 2 |y (t)| dt = 0 2 dt |x(t/T )| T =x CM (35.39) which completes the proof. Corollary 468 Schilders theorem remains true for Wiener processes on R+ , with good rate function J given by the eective Wiener action on R+ , J (x) 1 2 0 2 |x(t)| dt (35.40) if x H , J (x) = otherwise. Proof: For each natural number n, let n x be the restriction of x to the interval [0, n]. By Corollary 467, each of them obeys an LDP with rate function 2 1n 2 0 |x(t)| dt. Now apply the pro jective limit theorem (420) to get that J (x) = supn Jn (x), which is clearly Eq. 35.40, as the integrand is non-negative. CHAPTER 35. FREIDLIN-WENTZELL THEORY 35.2 244 Large Deviations for SDEs with State-Indep endent Noise Having established an LDP for the Wiener process, it is fairly straightforward to get an LDP for stochastic dierential equations where the driving noise is independent of the state of the diusion process. Denition 469 (SDE with Small State-Indep endent Noise) An SDE with small state-independent noise is a stochastic dierential equation of the form dX = a(X )dt + dW X (0) = 0 (35.41) (35.42) where a : Rd Rd is uniformly Lipschitz continuous. Notice that any non-random initial condition x0 can be handled by a simple change of coordinates. Denition 470 (Eective Action: State-Indep endent Noise) The eective action of a trajectory x H is J (x) 1 2 t 0 2 |x(t) a(x(t))| dt (35.43) and = if x C \ H . Lemma 471 The map F : C(R+ , Rd ) C(R+ , Rd ) given by t x(t) = w(t) + a(w(s))ds (35.44) 0 when x = F (w) is continuous. Proof: This goes rather in the same manner as the proof of existence and uniqueness for SDEs (Theorem 216). For any w1 , w2 C(R+ , Rd ), set x1 = F (w1 ), x2 = F (w2 ). From the Lipschitz property of a, |x1 (t) x2 (t)| w1 w2 + Ka t 0 |x1 (s) x2 (s)| ds (35.45) (writing |y (t)| for the norm of Euclidean vectors y , and x for the supremum norm of continuous curves). By Gronwalls Inequality (Lemma 214), then, x1 x2 w1 w2 eKa T (35.46) on every interval [0, T ]. So we can make sure that x1 x2 is less than any desired amount by making sure that w1 w2 is suciently small, and so F is continuous. 245 CHAPTER 35. FREIDLIN-WENTZELL THEORY Lemma 472 If w H , then x = F (w) is in H . Proof: Exercise 35.3. Theorem 473 (Freidlin-Wentzell Theorem: State-Indep endent Noise) The It processes X of Denition 469 obey the large deviations principle with o rate 2 and good rate function given by the eective action J (x). Proof: For every , X = F ( W ). Corollary 468 tells us that W obeys the large deviation principle with rate 2 and good rate function J . Since (Lemma 471) F is continuous, by the Contraction Principle (Theorem 410) X also obeys the LDP, with rate given by J (F 1 (x)). If F 1 (x) H = , this is . On the other hand, if F 1 (x) does contain curves in H , then J (F 1 (x)) = J (F 1 (x) H ). By Lemma 472, this implies that x H , too. For any curve w F 1 (x) H , x = w + a(x), or w = x a(x). 2 J (w) = 0 |x a(x)| dt is however the eective action of the tra jectory x (Denition 470). 35.3 Large Deviations for State-Dep endent Noise If the diusion term in the SDE does depend on the state of the process, one obtains a very similar LDP to the results in the previous section. However, the approach must be modied: the mapping from W to X , while still measurable, is no longer necessarily continuous, so we cant use the contraction principle as before. Denition 474 (SDE with Small State-Dep endent Noise) An SDE with small state-dependent noise is a stochastic dierential equation of the form dX = a(X )dt + b(X )dW X (0) = 0 (35.47) (35.48) where a and b are uniformly Lipschitz continuous, and b is non-singular. Denition 475 (Eective Action: State-Dep endent Noise) The eective action of a trajectory x H is given by J (x) where L(q , p) = L(x(t), x(t))dt 1 (pi ai (q )) Bij 1 (q ) (pj aj (q )) 2 and with J (x) = if x C \ H . (35.49) 0 B (q ) = b(q )bT (q ) (35.50) (35.51) CHAPTER 35. FREIDLIN-WENTZELL THEORY 246 Theorem 476 (Freidlin-Wentzell Theorem: State-Dep endent Noise) The processes X obey a large deviations principle with rate 2 and good rate function equal to the eective action. Proof: Considerably more complicated. (See, e.g., Dembo and Zeitouni (1998, sec. 5.6, pp. 213220).) The essence, however, is to consider an approximating It process Xn , where a(Xt ) and b(Xt ) are replaced in Eq. 35.47 by o a(Xn ([tn]/n)) and b(Xn ([tn]/n)). Here the mapping from W to Xn is continuous, so its not too hard to show that the latter obey an LDP with a reasonable rate function, and also that theyre exponentially equivalent (in n) to X . 35.4 Exercises Exercise 35.1 Prove Lemma 460. Exercise 35.2 Prove that Lemma 463 implies the large deviations lower bound for open sets. Exercise 35.3 Prove Lemma 472. Hint: Use Gronwal ls Inequality (Lemma 214) again to show that F maps HT into HT , and then show that H = n=1 Hn .
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Michigan - STAT - 36-754
BibliographyAbramowitz, Milton and Irene A. Stegun (eds.) (1964). Handbook of Mathematical Functions . Washington, D.C.: National Bureau of Standards. URLhttp:/www.math.sfu.ca/cbm/aands/.Algoet, Paul (1992). Universal Schemes for Prediction, Gambling a
Michigan - STAT - 36-754
Solution to Homework #1, 36-75427 January 2006Exercise 1.1 (The product -eld answers countable questions)Let D = S X S , where the union ranges over all countable subsets S of the index set T . For any event D D, whether or not asample path x D depend
Michigan - STAT - 36-754
Solution to Homework #2, 36-7547 February 2006Exercise 5.3 (The Logistic Map as a MeasurePreserving Transformation)The logistic map with a = 4 is a measure-preserving transformation, and the measure it preserves has the density 1/ x (1 x)(on the unit
Michigan - STAT - 36-754
Solution to Homework #3, 36-75425 February 2006Exercise 10.1I need one last revision of the denition of a Markov operator: a linear operatoron L1 satisfying the following conditions.1. If f 0 (-a.e.), then Kf 0 (-a.e.).2. If f M (-a.e.), then Kf M (
Michigan - STAT - 36-754
Syllabus for Advanced Probability II,Stochastic Processes36-754Cosma ShaliziSpring 2006This course is an advanced treatment of interdependent random variablesand random functions, with twin emphases on extending the limit theoremsof probability fro
George Mason - STAT - 344
Introduction to Engineering StatisticsLecture 02 TopicsCollecting engineering dataMechanistic and empirical modelsProbability and probability modelsLecture 02 Reference:Montgomery: Sec 1.2 through 1.41Basic Types of StudiesThree basic methods for
George Mason - STAT - 344
Probability ALecture 03 TopicsRandom experimentsSample spacesEventsCounting techniquesLecture 03 Reference:Montgomery: Sec 2.112ProbabilityCHAPTER OUTLINE2-1 Sample Spaces &amp; Events2-1.1 Random Experiments2-1.2 Sample Spaces2-1.3 Events2-1.
George Mason - STAT - 344
Probability BLecture 04 TopicsEqually likely outcomesProbability rulesUnions, intersections &amp; complementsSet operationsConditional probabilities in treesLecture 04 Reference:Montgomery:Sec 2.2 Axioms of ProbabilitySec 2.3 Addition rulesSec 2.4
George Mason - STAT - 344
Probability CLecture 05 TopicsMultiplication ruleTotal probability ruleIndependence of eventsReliabilityBayes TheoremRandom variablesLecture 05 Reference:Montgomery:Sec 2.5Sec 2.6Sec 2.7Sec 2.8Multiplication, total probability rulesIndepend
George Mason - STAT - 344
Discrete Probability ALecture 06 TopicsDiscrete random variables, defined &amp; graphedCumulative distribution functions, defined &amp;graphedMean and variance of a discrete random variableDefined mathematicallyGraphically explainedLecture 06 Reference:M
George Mason - STAT - 344
Discrete Probability BLecture 07 TopicsFor each of these distributions, we will examine the:Graph and parametersProbability mass and cumulative distribution functionsMean and varianceUniform distributionBinomial distribution:Negative binomial dist
George Mason - STAT - 344
Discrete Probability CLecture 08 TopicsFor each of these distributions, we will examine the:Graph and parametersProbability mass and cumulative distribution functionsMean and varianceHypergeometric distributionPoisson distributionLecture 08 Refere
George Mason - STAT - 344
Probability &amp; Statistics forEngineers/Scientists ILecture 01 TopicsIntroduction to the Syllabus, Assignment SheetBlackboard for course materials, lecture notesIntroduction to the instructorBasic ideas in statisticsIllustration of computer tools RL
George Mason - STAT - 344
Continuous Probability ALecture 09 TopicsContinuous variable distribution propertiesPDF &amp; CDF functions and graphsDerivation of the mean and varianceDesign and uses of the uniform distributionLecture 09 Reference:Montgomery:Sec 4.1Sec 4.2Sec 4.3
George Mason - STAT - 344
Continuous Probability BLecture 10 TopicsNormal distribution graphs and parametersStandard normal calculation, table and softwareApproximating discrete distributions with the normalExponential distributionFormula, graphs and parameterApplicationsL
George Mason - STAT - 344
Continuous Probability CLecture 11 TopicsBuilding on the exponential distribution of prior lectureMotivation, formula, graph, parameters andapplications of the:Erlang distribution and its extension, the gamma distributionWeibull distributionLognorm
George Mason - STAT - 344
Joint Probability Distributions ALecture 12 TopicsBuilding on the exponential distribution of prior lectureMotivation, formula, graph, parameters andapplications of the:Erlang distribution and its extension, the gamma distributionWeibull distributio
George Mason - STAT - 344
Joint Probability Distributions BLecture 13 TopicsPairwise independent random variablesRectangular ranges are necessary, but not sufficientFinding these probability distributions (&gt; 2 dimensions)Joint, marginal and conditional distributionsIndepende
George Mason - STAT - 344
Joint Probability Distributions CLecture 14 TopicsDiscrete multinomial distributionContinuous bivariate normal distributionIndependentDependent (covariance &amp; correlation)Reproductive propertyLinear combinations of random variablesSums and averages
George Mason - STAT - 344
General Bivariate Continuous DistributionsThis continuous variable example illustrates1) Finding the marginal and conditional for the two variables andcorresponding expected values, variances, and standarddeviations.2) Finding general conditional dis
George Mason - STAT - 344
Bivariate Discrete DistributionsLet X and Y be two discrete random variables defined on a samplespace S of an experiment.The joint probability mass function p(x, y) is defined for each pair ofnumbers (x, y) byIn this class the pairs of numbers can be
George Mason - STAT - 344
Gamma DistributionThe gamma distribution with parameters r and can be thought of asthe waiting time for r Poisson events when r is integer. The parameteris the expected number of Poisson events per a unit time interval. Ifincrease the typical wait for
George Mason - STAT - 344
Review:MarginalandConditionalDistributionsandCovarianceforContinuousDistributionsManytopicsinthetextbeginwithgeneralcaseexamplesandthencallattentiontofamiliesofdistribution,especiallythenormalfamily.Thefollowingusesapolynomialdensityfortworandomvariabl
George Mason - STAT - 344
Midterm 2 Overview by ChapterChapter 4 Continuous distributionsFamilies: Identification, domains, expected value variance: See SummaryProbability problems:R script: Normal Distribution, Exponential Distribution, Gamma DistributionHand integration: Si
George Mason - STAT - 344
1. Probability Density Functions from Chapter 4.In the Midterm exam, some density functions will be provided. You may be asked to fill in anyof the additional information: the family names, the domain possible values, and the expectedvalue and variance
George Mason - STAT - 344
Analysis of Paired DataThe Paired t TestThe sample consists of n independently selected items for which a pairof observations is made.We can compute the difference for each pairs and make inferencesabout the mean of these differences using a one samp
George Mason - STAT - 344
Data Type, Population Parameters and R Functionsfor Hypothesis Test and Confidence IntervalsSingle Population InferenceDataParameterR functionCount or fractionProportion pbinom.testof n itemsin class of interestContinuousMean t.testPaired co
George Mason - STAT - 344
Inference about a Difference BetweenPopulation ProportionsExample problem:Olestra was a fat substitute used in some snack foods.After some people consuming such snacks reported gastrointestinalproblems an experiment was performed.Results:90 of 563
George Mason - STAT - 344
Interpreting R Hypothesis Test and Confidence Interval OutputProblems are worth .5 points each. There are 50 problems.Directions: Most answers are very short. Round many digits answers to 2 significant digits.Write neatly giving the problem number and
George Mason - STAT - 344
Interpreting R Hypothesis Test OutputIn writing numeric values for answers, round to 3 significant digits.1.Exact binomial testdata: 12 and 24number of successes = 12, number of trials = 24, p-value = 0.03139alternative hypothesis: true probability
George Mason - STAT - 344
George Mason - STAT - 344
R Inputx = c( 25.8, 36.6, 26.3, 21.8, 27.2)t.test( x, alternative=&quot;greater&quot;, mu=25, conf.level=.95)R OutputOne Sample t-testdata: xt = 1.0382, df = 4, p-value = 0.1789alternative hypothesis: true mean is greater than 2595 percent confidence interv
George Mason - STAT - 344
Concepts of Point EstimationLecture 18 (former 17) Topics Basic properties of a confidence interval Large-sample confidence intervalsPopulation mean for measurement dataPopulation proportion for categorical data Bootstrap confidence intervals ignore
George Mason - STAT - 344
Confidence IntervalsLecture 20 TopicsVariancesProportionsPrediction intervalsLecture 19 Reference:Montgomery Sections 9-1 thru 9-3Devore Lecture 20Devore Lecture 211Hypothesis and Test ProceduresLecture 20 TopicsHypothesis tests versus confide
George Mason - STAT - 344
Risks and P-ValuesLecture 21and 22 TopicsType II errors risksP-ValuesLecture 21 Reference:Montgomery Sections 9-4, 9-1Excel WSReviewedStat 344 Lecture 221 RisksGo to file: Stat 344 Lecture 21 WSconcerning the interaction of theseinterrelate
George Mason - STAT - 344
dcfeae7461006edd771c0bf8ba9d38963497f08b.xlsDr. SimsIllustration of Defined Alternative HypothesisInput DataH0: =75H1: ==n==7491000.01Output DataIntermediate Calcs7070.470.871.271.67272.472.873.273.67474.474.875.275.67676.4
George Mason - STAT - 344
Two-Sample t-test proceduresTwo-sample t-test procedures enable inference about the difference ofmeans for two populations,Samples from the two populations denoted 1 and 2 are stored invectors called x and y for convenience.The procedures make use of
George Mason - STAT - 344
Tests concerning a population mean.The mean of a random sample from a population provides afoundation for creating a test statistic to assesses hypothesis about apopulation mean.Case 1. The population is from the normal family with meanThe standard d
George Mason - STAT - 344
Tests concerning a Population ProportionBackground: Large Sample TestsCommon large sample test statistics have form Z =.is the estimator for the population parameter of interest.is the expected value under the Null Hypothesis.is standard deviation o
George Mason - STAT - 344
Quiz1Scope ThisisaclosedbookandnotesquizrelatedtoChapter1and associatedRscripts. Thescopeisgivenbelow. Hopefullymanywillgetaperfectscope. 1. BeabletousewordstodescribedensityplotsasinFigure 1.11 2. Beabletowritethedefinitionsofthemeanandmedianon page25and
George Mason - MTH - 203
(J / jS O lUlIM ath 203-001 Spring 2011E xam 1Name: L astF irst( Problem 1 ) (25 points) F ind t he g eneral so lution o f t he linear s ystem (pleasewrite t he soluti on in t he v ector form) o r e xpla in w hy t he s ystem is inconsistent .- X2
George Mason - MTH - 203
~c7L ~T ()~JM a th 203-001 Spring 2011E xam 2N a rne: LastF irst( Prob le m 1) ( 18 point s) C ompute t h e fo llowi ng determin a nt s. Show s teps b u t tryt o avoid u nn ecessary c alcul at ions when possible.2o51-1 3237-644L il@68-
George Mason - MTH - 203
S OL U I) OJ\!M ath 203-001 Spring 2011E xam 3F irstName: L ast(P roblem 1 ) (25 points) For t he m atrix A =[~ ~]do t he following:(1) F ind all eigenvalues;(2) For each eigenvalue, find t he basis of t he eigenspace;(3) I f i t t urns o ut t h
Punjab Engineering College - LALA - 222
Clayton VHS AP Physics B 05-06 Chapter 4 Quiz SolutionsClayton VHS AP Physics B 05-06 Chapter 4 Quiz SolutionsClayton VHS AP Physics B 05-06 Chapter 4 Quiz SolutionsClayton VHS AP Physics B 05-06 Chapter 4 Quiz SolutionsClayton VHS AP Physics B 05-06
Punjab Engineering College - LALA - 222
Clayton VHS AP Physics B 05-03 Chapter 4 Homework SolutionsClayton VHS AP Physics B 05-03 Chapter 4 Homework SolutionsClayton VHS AP Physics B 05-03 Chapter 4 Homework SolutionsClayton VHS AP Physics B 05-03 Chapter 4 Homework SolutionsClayton VHS AP
Punjab Engineering College - LALA - 222
Punjab Engineering College - LALA - 222
Amber PatelWith the increase of the rate of heart diseases the necessity for artificial hearts has increasedrapidly. Heart diseases are the highest killers in the world ranging from heart attack to totalstoppage of the heart. With the demand of artific
Punjab Engineering College - LALA - 222
Amber PatelCollisionsPurpose: To use law of conservation of energy to find the ratio of mass of two colliding objects.Procedure:To do this we are going to watch three videos of collisions of cars on air tracks. The air track hasalmost zero friction s
Punjab Engineering College - LALA - 222
Amber PatelCollisionsPurpose: To use law of conservation of energy to find th e ratio of mass of two colliding objects.Procedure:To do this we are going to watch three videos of collision s of cars on air tracks. The air track hasalmost zero friction
Punjab Engineering College - LALA - 222
Week 6: Conceptual Questions Chapter 7 Homework:#3: An object executes circular motion with a constant speedwhenever a net force of constant magnitude acts perpendicular to itsvelocity. What happens to the speed if the force is not perpendicularto the
Punjab Engineering College - LALA - 222
Punjab Engineering College - LALA - 222
Punjab Engineering College - LALA - 222
Punjab Engineering College - LALA - 222
Punjab Engineering College - LALA - 222
Punjab Engineering College - LALA - 222
Punjab Engineering College - LALA - 222
Amber PatelCar LabWeight of car: 3264lb FR: 1684 RR: 1580Observation:TireArea ofWeight2Contact(in )Supportedby Tire(lb) [FR:1684/2RR:1580/2]Left Front32.4842Right Front 32.8842Left Rear30.2790Right Rear 30.5790PressureCalculated
Punjab Engineering College - LALA - 222
Punjab Engineering College - LALA - 222