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
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