Good Review Materials. Random Variables and Random Vectors. Random variables. Discrete random variables. Continuous random variables
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1 Good Rvw Matrals Radom Varabls ad Radom Vctors ttp:// (Gozals & Woods rvw matrals Capt. : Lar Algbra Rvw Capt. 2: Probablty, Radom Varabls, Radom Vctors Radom varabls Dscrt radom varabls Sampls from a radom varabl ar ral umbrs A radom varabl s assocatd wt a probablty dstrbuto ovr ts ral valus Two typs of radom varabls Dscrt Oly ftly may possbl valus for t radom varabl: X {a, a 2,, a } (Could also av a coutabl fty of possbl valus».g., t radom varabl could tak ay postv tgr valu Eac possbl valu as a ft probablty of occurrg. Cotuous Iftly may possbl valus for t radom varabl E.g., X {Ral umbrs} Dscrt radom varabls av a pmf (probablty mass fucto, P P(X = a = P(a Eampl: Co flp P (a X = 0 f ads X = f tals Wat s t pmf of ts radom varabl? 0 0 a Dscrt radom varabls Cotuous radom varabls Dscrt radom varabls av a pmf (probablty mass fucto, P P(X = a = P(a P (a Eampl: D roll X {, 2, 3, 4, 5, 6} Wat s t pmf of ts radom varabl? a Cotuous radom varabls av a pdf (probablty dsty fucto, p Eampl: Uform dstrbuto p( 0 2 p(.3 =? p(2.4 =? Wat s t probablty tat X =.3 actly: P(X =.3 =? Probablty corrspods to ara udr t pdf..5 P( < X <.5 =? p(d = 0.25
2 Cotuous radom varabls Wat s t total ara udr ay pdf? p( p ( d =? Eampl cotuous radom varabl: Huma gts p( p( Radom varabls How muc cag do you av o you? Askg a prso (cos at radom tat qusto ca b tougt of as samplg from a radom varabl. Is t radom varabl Amout of cag popl carry dscrt or cotuous? Radom varabls: Ma & Varac A mportat typ of radom varabl Ts formulas ca b usd to fd t ma ad varac of a radom varabl w ts tru probablty dstrbuto s kow. Ma µ Dfto Dscrt r.v. Cotuous r.v. µ = E(X µ = a P( a µ = p(d Varac Var(X E(X ( µ 2 ( a µ 2 P(a ( µ 2 p( d T Gaussa dstrbuto Estmatg t Ma & Varac p( = 2π σ ( µ2 2σ 2 X ~ N(µ, σ 2 Aftr samplg from a radom varabl tms, ts formulas ca b usd to stmat t ma ad varac of t radom varabl. Sampls,, 3,, Estmatd ma: Estmatd varac: m = σ 2 = σ 2 = ( m 2 mamum lklood stmat ( m 2 ubasd stmat
3 Fdg ma, varac Matlab Sampls Ma = [ 3 ] >> m = (/*sum( Varac m σ 2 = [ m L m] m 2 M m Mtod : Mtod 2: >> v = (/*(-m*(-m >> z = -m >> v = (/*z*z Eampl cotuous radom varabl Popl s gts (mad up Gaussa µ = 67, σ 2 = 20 Wat f you wt to a plat wr gts Gaussa µ = 75, σ 2 = 5 How would ty b dffrt from us? Eampl cotuous radom varabl Tm popl wok up ts morg Gaussa µ = 9, σ 2 = Radom vctors A -dmsoal radom vctor cossts of radom varabls tat ar all assocatd wt t sam vts. Eampl 2-D radom vctor: wr X s radom varabl of uma gts V = X Y s radom varabl of wak-up tms Sampl tms from V. Lt s collct som v L v sampls ad grap tm: y L y (wak-up tms (gts Radom Vctors Ma of a radom vctor Wat wll t grap of V look lk? Estmatg t ma of a radom vctor sampls from V v L v L y Wat s ma of V? Ma of X s 67 Ma of Y s 0 m = 67 0 Ma m = v = To stmat ma of V Matlab >> (/*sum(v,2 y = m m y
4 Radom vctor Eampl 2-D radom vctor: wr X s radom varabl of uma gts V = X Y s radom varabl of uma wgts Sampl tms from V v L v Wat wll grap look lk? L y Covarac of two radom varabls Hgt ad wak-up tm ar ucorrlatd, but gt ad wgt ar corrlatd. Covarac Cov(X, Y = 0 for X = gt, Y = wak-up tms Cov(X, Y > 0 for X = gt, Y = wgt Dfto: Cov(X,Y = E(X µ (Y µ y ( If Cov(X, Y < 0 for two radom varabls X, Y, wat would a scattrplot of sampls from X, Y look lk? Estmatg covarac from sampls Sampl tms: L y Cov(X,Y = ( (y Cov(X,Y = ( (y Cov(X, X =?Var(X mamum lklood stmat ubasd stmat How ar Cov(X, Y ad Cov(Y, X rlatd? Cov(X, Y = Cov(Y, X Estmatg covarac Matlab Sampls Mas = [ 3 ] m m_ y = [y y 3 L y ] m y m_y Covarac y Cov(X,Y = [ m ] M y Mtod : >> v = (/*(-m_*(y-m_y Mtod 2: >> w = -m_ >> z = y-m_y >> v = (/*w*z Covarac matr of a D-dmsoal radom vctor I 2 dmsos V = X Cov(V = E V µ ( ( V µ T = E X µ X X µ X Y µ Y [ ] I D dmsos Cov(V = E( V µ ( V µ T W s a covarac matr symmtrc? A. always, B. somtms, or C. vr Var(X Cov(X,Y Y µ Y = Cov(X,Y Var(Y Eampl covarac matr Popl s gts (mad up X ~ N(67, 20 Tm popl wok up ts morg Y ~ N(9, Wat s t covarac matr of? V = X
5 Estmatg t covarac matr from sampls (cludg Matlab cod Sampl tms ad fd ma of sampls v L v V = m = m L y m y Fd t covarac matr Cov(V = L y M >> m = (/*sum(v,2 >> z = v - rpmat(m,, >> v = (/*z*z y M y Gaussa dstrbuto D dmsos -dmsoal Gaussa s compltly dtrmd by ts ma, µ, ad varac, σ 2 : p( = ( µ2 X ~ N(µ, σ 2 2σ 2π σ 2 D-dmsoal Gaussa s compltly dtrmd by ts ma, µ, ad covarac matr, Σ : p( = 2 ( µ T Σ ( µ X ~ N(µ, Σ (2π D /2 Σ /2 Wat apps w D = t Gaussa formula?
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