Introduction to Random Variables

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1 Introducton to Random Varables Defnton of random varable Defnton of random varable Dscrete and contnuous random varable Probablty functon Dstrbuton functon Densty functon Sometmes, t s not enough to descrbe all possble results of an eperment: Toss a con 3 tmes: {(HHH), (HHT), } Throw a dce twce: {(,), (,), (,3), } Some tne t s useful to assocate a number to each result of an eperment 3 Characterstc measures of a random varable Defne a varable Mean, varance Other measures We don t know the result of the eperment before we carry t out We don t know the value of the varable before the eperment 4 Transformaton of random varables Defnton of random varable Defnton of random varable Sometmes, t s not enough to descrbe all possble results of an eperment: A random varable s a functon whch assocates a real number to each element of the sample space Toss a con 3 tmes: {(HHH), (HHT), } Throw a dce twce: {(,), (,), (,3), } A veces es útl asocar un número a cada resultado del epermento. X = Number of head on the frst toss X[(HHH)]=, X[(THT)]=, Random Varables are represented n captal letters, generally the last letters of the alphabet: X,Y, Z, etc. No conocemos el resultado del epermento antes de realzarlo Y = Sum of ponts Y[(,)]=, Y[(,)]=3, The values taken by the varable are represented by small letters, No conocemos el valor que va a tomar la varable antes del epermento = s a possble value of X y=3. s a possble value of Y z=-7.3 s a possble value of Z 3 4

2 Defnton of random varable Defnton of random varable s E X(s ) = b; s E s k s Number of defectve unts n a random sample of 5 unts X(s k ) = a Number of faults per cm of materal Lfetme of a lamp a b R X Resstance to compresson of concrete The space R X s the set of ALL possble values of X(s). Each possble event of E has an assocated value n R X We can consder R as another random space 5 6 Defnton of random varable Introducton to Random Varables E X(s ) = b; s E s k s Defnton of random varable X(s k ) = a a The elements n E have a probablty dstrbuton, ths dstrbuton s also assocated to the values of the varable X. That s, all r.v. preserve the probablty structure of the random eperment that generates t: Pr( X = ) = Pr( s E: X( s) = ) b R X Dscrete Dscrete and and contnuos contnuous random random varables varable Probablty functon Dstrbuton functon Densty functon 3 Characterstc measures of a random varable Mean, varance Other measures 7 4 Transformaton of random varables 8

3 Dscrete and contnuous random varables Dscrete and contnuous random varables The rank of a random varable una varable aleatora s the set of possble values taken by the varable. Dependng on the rank, the varables can be classfed as: Dscrete: Those that that take take a fnte or or nfnte (numerable) number of of values Contnuous: Those whose rank rank s s an an nterval of of real real numbers s of of dscrete random varables Number of of faults on on a glass surface Proporton of of default parts n n a sample of of Number of of bts bts transmted and and receved correctly s of of contnuous random varables Electrc current Generally count the number of tmes that somethng happens Longtude Temperature Generally measure a magntude 9 Weght Dscrete random varables Dscrete random varables The values taken by a random varable change from one eperment to another, snce the results of the eperment are dfferent A r.v. s defned by The values that t takes. The probablty of takng each value. The propertes of the probablty functon come from the aoms of probablty:. P(A). P(E)= 3. P(AUB)=P(A)+P(B) s A B=Ø p ( ) p( ) n p ( ) = = { } { } a< b< c A= a X b B= b< X c Pr( a X c) = Pr( a X b) + Pr( b< X c) Ths s a functon that ndcates the probablty of each possble value p ( ) = PX ( = ) n

4 Dscrete random varables Eperment: Toss cons. X=Number of tals. Dscrete random varables Eperment: Toss cons. X=Number of tals. X E HH TH HT TT R X /4 / Pr H H T T H T H T X P(X=) /4 / /4 3 4 Dscrete random varables Dscrete random varables Eperment: Toss cons. X=Number of tals. Sometmes we mght be nterested on the probablty that a varable takes a value less or equal to a quantty p() X P(X=) = = = X /4 / /4 5 F( ) = P( X ) F( ) = F( ) = f X takes values K : n F( ) = P( X ) = p( ) F( ) = P( X ) = p( ) + p( ) F( ) = P( X ) = p( ) = n n = M n 6

5 Dscrete random varables Eperment: Toss cons. X=Number of tals. Dscrete random varables Eperment: Toss cons. X=Number of tals. p() X P(X=) F() X F() /4 /4 / / /4 = = = X = = = X 7 8 Contnuous random varables Contnuous random varables When a random varable s contnuous, t doesn t make sense to sum: = p ( ) = Densty functon descrbes the probablty dstrbuton of a contnuous random varable. It s a functon that satsfes: Snce the set of of values taken by the varable s not numerable We can generalze We ntroduce a new concept nstead of the probablty functon of dscrete random varables f( ) f( ) d= Pa ( X b) = f( ) d b a 9

6 Contnuous random varables Contnuous random varables Densty functon descrbes the probablty dstrbuton of a contnuous random varable. It s a functon that satsfes: f( ) PX ( = a) = f( ) d= a a f( ) d= Pa ( X b) = f( ) d b a a b Area below the curve P( a X b) = P( a< X b) = Pa ( X< b) = Pa ( < X< b) a Contnuous random varables Contnuous random varables The densty functon doesn t have to be symmetrc, or be defne for all values the form of the curve wll depend on one or more parameters y fx ( β ) 3 If we measure a contnuous varable and represent the values n a hstogram: If we make the ntervals smaller and smaller: 4

7 Contnuous random varables Contnuous random varables f ( ) 5 6 Contnuous random varables Contnuous random varables The densty functon of the use of a machne n a year (n hours ): What s the probablty that a machne randomly selected has been used less than 3 hours? f() f().4,.5.4 f () =.8,.5, < <.5.5 < 5 elsewhere P( X < 3. ) = = = d d

8 Contnuous random varables Contnuous random varables As n the case of dscrete random varables, we can defne the dstrbuton of a contnuous random varables by means of the Dstrbuton functon: As n the case of dscrete random varables, we can defne the dstrbuton of a contnuous random varables by means of the Dstrbuton functon: F( ) = P( X ) = f( u) du < < F( ) = P( X ) = f( u) du < < PX ( ) In the dscrete case, the Probablty functon s obtaned as the dfference of to adjon values of F(). In the case of contnuous varables: df( ) f( ) = d 9 3 Contnuous random varables Contnuous random varables The Dstrbuton functon satsfes the followng propertes: The Dstrbuton functon satsfes the followng propertes: a< b F( a) F( b) It s non-decreasng F( ) = F( ) = It s rght-contnuous a< b F( a) F( b) F( ) = F( ) = If we defne the followng dsjont events: { X a} { a< X b} { X a} { a< X b} = { X b} Pr( X b) = Pr( X a) + Pr( a< X b) F( b) Frst aom of probablty F( ) = Pr( X ) = f( ) d= F( ) = Pr( X ) = f( ) d= Thrd aom of probablty 3 3

9 Contnuous random varables Contnuous random varables The densty functon of the use of a machne n a year (en horas ):.4,.5.4 f () =.8,.5, < <.5.5 < 5 elsewhere.4 f() , < < f( ) =.8,.5 < 5.5, elsewhere Pr( < X <.5) Pr(.5 X < ).4 udu < < F( ) = u du+.8 u du,.5 < Pr( X 5) 5 34 Contnuous random varables Contnuous random varables P(<3.) P(<3.) =3. < < F = + < ( )

10 Introducton to Random Varables 3 Characterstc measures of a r.v. Central measures Defnton of random varable Dscrete and contnuous random varable Probablty functon Dstrbuton functon Densty functon 3 Characterstc measuresof of a random varable Mean, varance Other measures 4 Transformaton of random varables 37 In the case of a sample of data, the sample mean allocates a weght of /n to each value: = + + K+ n n n n The mean μ or Epectaton of a r.v. uses the probablty as a weght: [ ] μ = E X = [ ] μ = E X = p( ) f( ) d dscrete r.v. contnuous r.v Characterstc measures of a r.v. 3 Characterstc measures of a r.v. Central measures What s the average tme of use of the machnes? Intutvely: Medan = value that dvdes the total probablty n to parts PX ( m) =.5 Fm ( ) , < < f( ) =.8,.5 < 5.5, elsewhere E[ X] = f( ) d= d.8 d =.5 4

11 3 Characterstc measures of a r.v. 3 Characterstc measures of a r.v. Other measures If we want to know the tme of use such that 5% of the machnes have a use less or equal to that value Fm= ( ).5 The percentl p of a random varable s the value p that satsfes: < < F( ) = < =.5 m= =.5 m= p( X < ) p y p( X ) p F( ) = p p p p dscrete r.v. contnuous r.v. A specal case are quartles whch dvde the dstrbuton n 4 parts Q = p.5 Q = p = Medan Q.5 = p Characterstc measures of a r.v. 3 Characterstc measures of a r.v. Medsures of dsperson Medsures of dsperson [ ] = ( [ ]) Var X E X E X The sample varance of a set of data s gven by: s = ( ) + ( ) + K+ ( ) n n n n [ ] = ( [ ]) Var X E X E X [ ] = ( [ ]) Var X E X E X The Varance of a r.v. also uses the probablty as a weght: σ [ ] = Var X = ( μ) p( ) [ ] σ = Var X = ( μ) f ( ) d dscrete r.v. contnuous r.v. 43 ( [ ]) = + ( [ ]) [ ] E X E X E X E X XE X = E X + ( E[ X] ) E[ X] E[ X] E[ X] s a constant, does not depend on X = E X ( E[ X] ) It s a lnear operator 44

12 Introducton to Random Varables 4 Transformaton of random varables Defnton of random varable Dscrete and contnuous random varable In some stuatons we wll need to know the probablty dstrbuton of a transformaton of a random varable s Probablty functon Dstrbuton functon Densty functon 3 Characterstc measures of a random varable Change unts Use logarthmc scale sn X X ns ax + b X Mean, varance Other measures 4 4 Transformaton of of random randomvarables 45 X X log X Y = g(x ) X e X X 46 4 Transformaton of random varables 4 Transformaton of random varables Let X be a r.v. If we change to Y=h(X), we obtan a new r.v.: A company packs mcrochps n lots. It s know that the probablty dstrbuton of the number of mcrochps per lots s gven by: F ( y) = Pr( Y y) = Pr( h( X) y) = Pr( A) Y Dstrbuton functon Y = h( X) {, ( ) } A = h y p() F() Pr( X 44)? ( X ) { } X = A A= Pr( 44) Pr( ), 44 Pr =.6 {, 44} A= 48

13 4 Transformaton of random varables In general: Y = h( X) If h s contnuous and monotonc ncreasng : F y hx y X h y F h y ( ) Pr( ( ) ) Pr( Y = = ( )) = X( ( )) 4 Transformaton of random varables Densty functon If X s a contnuous r.v. Y=h(X), where h s dervable and nyectve d fy( y) = fx( ) dy If h s contnuous and monotonc decreasng: F y hx y X h y F h y ( ) Pr( ( ) ) Pr( Y = = ( )) = X( ( )) f Y FX ( ) d FY ( y) F ( h ( y)) d dy ( y) = = = y y ( FX ( )) d d dy ncreasng decreasng Transformaton of random varables 4 Transformaton of random varables If X s a contnuous r.v. Y=h(X), where h s dervable and nyectve The velocty of a gas partcle s a r.v. V wth densty functon d fy( y) = fx( ) dy f V () v = bv ( b /) v e v> elsewhere For dscrete r.v.: The knetc energy of the partcle s functon of W? W = mv / What s the densty p ( y) = Pr( Y = y) = Pr( X = ) Y h( ) = y 5 5

14 4 Transformaton of random varables The velocty of a gas partcle s a r.v. V wth densty functon 4 Transformaton of random varables The velocty of a gas partcle s a r.v. V wth densty functon f V () v = bv ( b /) v e v> elsewhere f V () v = bv ( b /) v e v> elsewhere W = mv / v= w/ m v= w/ m dv dw mw = ( ) f ( h ( w)) = ( b /) w/ m e V b w/ m f W ( w) = b w/ m ( b / m) w/ m e w> elsewhere Transformaton of random varables 4 Transformaton of random varables Epectaton Epectaton h ( ) fx ( d ) E[ h( X) ] = h ( ) px ( = ) Y = h( X), h( ) = y ncreasng d E[ y] = yfy( y) dy = h( ) fx( ) dy dy [ ( )] E h X =, h( ) = y h ( ) f ( d ) X h ( ) px ( = ) Lnear Transformatons Y = a+ bx 55 [ ] = + [ ] [ ] = b Var [ X ] EY a be X Var Y 56

A random variable is a function which associates a real number to each element of the sample space

A random variable is a function which associates a real number to each element of the sample space Introducton to Random Varables Defnton of random varable Defnton of of random varable Dscrete and contnuous random varable Probablty blt functon Dstrbuton functon Densty functon Sometmes, t s not enough

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