Statistics and Probability Theory in Civil, Surveying and Environmental Engineering

Transcription

1 Statstcs and Probablty Theory n Cvl, Surveyng and Envronmental Engneerng Pro. Dr. Mchael Havbro Faber ETH Zurch, Swtzerland

2 Contents o Todays Lecture Overvew o Uncertanty Modelng Random Varables - propertes o the expectaton operator - random vectors and jont moments - condtonal dstrbutons and condtonal moments - the probablty dstrbuton or the sum o two random varables - the probablty dstrbuton or unctons o random varables

3 Overvew o Uncertanty Modelng Random varables and ther characterstcs Real world Model Uncertan phenomenon Random varables Data/observatons x,y p(x,y), , , , , ,0 0.00, , , , , , , , , ,

4 Propertes o the expectaton operator The expectaton operator acltates that we can assess the expected value and the varance o a random varable By understandng how the expectaton operator works we wll be able to assess the expected value and the varance o unctons o random varables Ths s useul we want to analyze engneerng models nvolvng one or more random varables n regard to ther expected values and ther varances E.g.: Duraton o a constructon process as a uncton o the duraton o ts ndvdual processes

5 Propertes o the expectaton operator The expectaton operator possesses the ollowng propertes: E E [] = c [ ] = ce[ ] [ a + b ] = a + be[ ] [ g ( ) + g ( )] = E[ g ( )] E[ g ( )] E c E c +

6 Propertes o the expectaton operator The varance can thus be wrtten as: [ ] Var = E ( μ ) = E + μ μ [ ] = μ + E μe = μ + E μ = E μ

7 Propertes o the expectaton operator Furthermore there s Var Var Var [] c = 0 [ ] c = c Var[ ] [ ] a + b = b Var[ ] E E [ ] = c [ ] = ce[ ] [ a + b ] = a + be[ ] [ g ( ) + g ( )] = E[ g ( )] E[ g ( )] E c E c +

8 Propertes o the expectaton operator From the result [ ] = ( μ ) = + μ μ = μ Var E E E t s seen that there n general s E [ g( ) ] g( E[ ]) E [ g( )] g( E[ ]) or convex unctons - Jensen s nequalty! Equalty only or lnear unctons

9 Random vectors and jont moments Oten we are dealng wth models nvolvng not only one random varable but several random varables These random varables can be collected n a vector In general the components o the vector are dependent E.g. Ranall and water level It s thus necessary that we establsh probablstc models whch nclude ths dependency we can do ths through the jont cumulatve dstrbutons and the jont moments.

10 Random vectors and jont moments Now we consder not just one contnuous random varable but a vector o contnuous random varables = (,,, ) n T The jont cumulatve dstrbuton uncton s gven by ( ) = ( ) F x P x x x n n and the jont probablty densty uncton s gven by n z z z ( x) = F ( x) n

11 Random vectors and jont moments Consder the two dmensonal dscrete probablty densty uncton: x,y p(x,y), , , , , ,0 0.00, , , , , , , , , ,

12 Random vectors and jont moments The margnal probablty densty uncton o a random varable s dened by ( ) = ( ) x ( n old) dx.. dx dx.. dx x + n

13 Random vectors and jont moments Consder the two dmensonal dscrete probablty densty uncton: x,y p(x,y), , , , , ,0 0.00, , , , , , , , , , Dscrete jont densty Margnal densty or x 40 0

14 Random vectors and jont moments C C The covarance between the th and the j th component o the random vector o contnuous random varables s dened as the jont central moment.e. by j [ ] ( )( ) ( ) μ )( j μ ) = x μ x j x,x j dx dxj = E( μ [ ] = Var j j j From where we see that or = j we get the varance or Correlaton coecent ρ j C j = = σ σ j ρ

15 Random vectors and jont moments The expected value and the varance o a lnear uncton n = + 0 = Y a a are gven by [ ] = + [ ] EY a ae 0 n n = [ ] [ ] Var Y = a Var + a a C j =, j= j n j

16 Condtonal dstrbutons and condtonal moments Some tmes t s useul to be able to assess the probablty o an event gven that we know somethng about one o the random varables whch are used to dene the event E.g. assume we want to calculate the probablty that a project wll be delayed under the condton that one o the processes wll exceed ts planned duraton by 50%.

17 Condtonal dstrbutons and condtonal moments The condtonal probablty densty uncton or the random varable gven the outcome o the random varable s gven by ( x x ) = ( x where and are ndependent The condtonal cumulatve dstrbuton uncton s obtaned by ntegraton as x (, zx) dz, F ( x x) = ( x ), ( x,x ( x x ) = ( x ) ) )

18 Condtonal dstrbutons and condtonal moments The un-condtonal cumulatve dstrbuton uncton or the random varable can be derved rom the condtonal comulatve dstrbuton uncton by use o the total probablty theorem F ( x ) = F ( x x ) ( x ) dx μ The condtonal expected value s dened by ( ) = E = x = x x x dx

19 In many cases we are nterested n assessng the probabltes o unctons o random varables The unctons are useul or descrbng the events we are nterested n they are our engneerng models. A smple case s the sum o two random varables t s useul to derve the cumulatve dstrbuton uncton or such a sum. A more general case concerns monotonc unctons o random varables we wll also derve the cumulatve dstrbuton or ths case.

20 The cumulatve dstrbuton uncton or the sum o two random varables Consder the sum Y = + and assume that we have, ( x, x ) Frst we derve the densty uncton or Y = x + assumng that s gven.e. ( yx) = ( y x x) Y ( x x ) =, ( x ( x,x ) ) and we get (, y x) = ( y x x) ( x) = ( y x, x) Y,,

21 The cumulatve dstrbuton uncton or the sum o two random varables The margnal probablty densty uncton or Y s now acheved by ntegratng out over,.e. Y ( y ) =, ( y x,x ) dx For thecasewhere and are ndependent we get the so-called convoluton ntegral Y ( y ) = ( y x ) ( x ) dx

22 The cumulatve dstrbuton uncton or unctons o random varables Consder the more general problem o dervng the cumulatve dstrbuton uncton or a uncton o a random varables.e. Y = g( ) where the probablty dstrbuton uncton o s gven as F () x I g() x s monotoncally ncreasng and represents a one-to-one mappng, a realzaton o Y s only smaller than y 0 the realzaton o s smaller than x 0 where F y PY y P g y ( ) ( ) ( ( )) Y = = The cumulatve dstrbuton uncton or Y s then gven by x = g ( y ) 0 0 F y F g y Y( ) = ( ( ))

23 The cumulatve dstrbuton uncton or unctons o random varables startng now wth F y F g y Y( ) = ( ( )) we have () ( F ( )) g y Y y = y x Y( y) = g ( y) ( g ( y)) ( ) ( ) Y y = x y y

24 The cumulatve dstrbuton uncton or unctons o random varables In case the uncton g() x s monotoncally decreasng, a realzaton o Y s only smaller than y 0 the realzaton o s larger than x 0, and n ths case we have to change the sgn.e. F y F g y Y( ) = ( ( )) yeldng x Y( y) = ( x) y In the general case or monotoncally ncreasng or decreasng unctons there s thus x Y( y) = ( x) y

25 The cumulatve dstrbuton uncton or unctons o random varables For thecasewherethecomponentso a randomvectory= ( YY,,.. Y) T n can be gven as one-to-one mappngs o monotoncally ncreasng or decreasng unctons g, =,,.. n o the components o a random vector = (,,.. ) T n n the orm: Y = g ( ) there s wth J Y () y = J () x x x beng the absolute value o the determnant o... y y n J = xn xn... y n