Uncertainty Modelling M. H. Faber. Yovisto Academic Video Search. Probability functions, The central limit theorem, The Normal distribution, The Lognormal distribution, Stochastic processes, Random sequences, Bernoulli trials Random Variables Stochastic Processes Normal Extremes Uncertainty Small Modeling distribution Theorem Overview Central Tools SEQUENCES ETH Zurich stochastic process extrem apply this result chanc getting requir toss with dic experiencing earthquak annual probability year swiss federal institut technology process extrem median geometric distribution provid information regard long need play gam with probability winning tim unit tak natural logarithm both sid that swiss federal institut technology stochastic stochastic process extrem median geometric distribution provid information regard long need play gam with probability winning tim unit ord hav chanc unit might toss dic year earthquak defined through determin function swiss federal institut technology exampl easily thus approach applied probl deta swiss federal institut technology exampl rememb that could establish function vector random wer giv function variabl swiss federal institut technology stochastic process extrem random sequenc probability density function numb independent trial befor first success giv corresponding distribution thus ility swiss federal institut technology stochastic process extrem expected valu varianc binomially distributed random variabl giv swiss federal institut technology function exampl rememb that probability density random variabl swiss federal institut technology sequenc experiment with only possibl mutually exclusiv outcom called bernoulli trial binomial probability density function swiss federal institut technology stochastic process extrem sequenc stochastic process extrem sequenc sequenc experiment with only possibl mutually exclusiv outcom called bernoulli trial binomial probability distribution function then follows swiss federal institut technology stochastic process extrem random sequenc sequenc experiment with only possibl mutually exclusiv outcom called bernoulli trial typically trial denoted success failur probability success constant equal density giv ii-a binomial operator function swiss federal institut technology stochastic process extrem sequenc sequenc experiment with only possibl mutually exclusiv outcom called bernoulli trial binomial probability distribution function then follows swiss federal institut technology stochastic process extrem random sequenc sequenc experiment with only possibl mutually exclusiv outcom called bernoulli trial typically trial denoted success failur probability success constant equal density giv binomial operator function swiss federal institut technology process wind velocity wav height swiss federal institut technology stochastic process extrem random quantiti tim variant sens that they tak valu different tim trial realization occur discret hav quantity called sequenc failur event traffic congestion continuously continu exampl rememb convolution integral which used establishing density function random flirt thus work independent uniformly distributed variabl swiss federal institut technology exercis expectation operator rewrit lik this swiss federal institut technology random variabl ther exist larg numb different cumulativ probability uniform exponential beta swiss federal random variabl normal cas wher mean valu equal zero standard deviation variabl said numb at-i swiss federal institut technology random variabl when logarithm variabl normal distributed said swiss federal institut technology cxli riv oft used model uncertain paramet which cannot hav negativ realization fatigu lif steel concret resistanc daily flows whenev variabl result product several swiss federal institut technology random variabl log-normal random variabl concret compression strength probability valu low than swiss federal institut technology random variabl log-normal distribution oft used model uncertain paramet which cannot hav negativ realization fatigu lif steel concret resistanc daily riv flows whenev variabl result product several swiss federal institut technology random variabl log-normal distribution useful property that independent distributed with paramet then paramet swiss federal institut technology random variabl when logarithm variabl normal distributed said swiss federal institut technology random variabl wher normal follows from central limit theor log-normal distribution product swiss federal institut technology ajli jwst overview uncertainty modeling random their del real worl chat overview uncertainty modeling random variabl their characteristics model reat real worl swiss federal institut technology what probability that measurement actual numb giv random counted perfect swiss federal institut technology exercis exercise1 vhat that measurement actual numb counted perfect nted unstalled swiss federal institut technology exercis villag traffic passing through cent being counted installed devic control correct function this very costy perfectly counting used flow measured both devic shown joint probability density below what that inprecis measurement show actual numb giv swiss federal institut technology exercise1 that measurement joint acut gwen actual o-ii evil numb counted perfect swiss federal institut technology altt random variabl normal distribution accumulation eith left right each eight lay hrw4 tun bit random movement eith left right ight random normal distribution accumulation eith left right each eight lay random variabl normal distribution accumulation movement eith left right each eight lay swiss federal institut technology normal distribution result from oth consideration energy isolated noon closed lady swiss federal institut technology random variabl random variabl normal distribution analytical form derived repeated result regarding probability density function very frequently applied engineering modeling when quantity assumed composed numb individual contribution linear combination distributed thus variabl swiss federal institut technology random variabl central limit theor stat numb approach normal becom larg swiss federal institut technology random variabl central theor condition validity should dominated component statistical dependency betwe requirement typ hav skew distrubution numb swiss federal institut technology random variabl illustration swiss federal institut technology random variabl illustrat structural memb rul length small unit measurement rounded clos each measurement subject uncertainty rang accumulated error associated with over length betwe four aught meos rement swiss federal institut technology central limit theor validity should dominated component dependency betwe requirement typ distribution hav skew numb swiss federal institut technology random variabl random variabl central theor stat function numb approach normal becom larg swiss federal institut technology content todays lectur random variabl central limit theor normal distribution log-normal stochastic process extrem sequenc bernoulli trial binomial geometric swiss federal institut technology tool uncertainty modeling random variabl them characteristics function function swiss federal institut technology tool uncertainty modeling random them expectation operator jointly distributed variabl eatr mad swiss federal institut technology tool uncertainty modeling patt helps identify usefulness solution strategi from problem swiss federal institut technology tool uncertainty modeling problem thos involving very oft uniqu abl solv requir basuc physical natural human scienc engineering being identify ways solving training important becaus provid experienc traming start recogniz pattern swiss federal institut technology overview uncertainty modeling random variabl their characteristics fall swiss federal institut technology technology overview uncertainty modeling random variabl their design galleri swiss federal institut overview uncertainty modeling random their characteristics model real worl swiss federal institut technology basic statistics probability theory civil surveying environmental engineering prof michael fab swiss federal institut technology zurich switzerland