Pairs of Random Variables
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- Μελέτη Γεωργιάδης
- 6 χρόνια πριν
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1 Pairs of Random Variabls Rading: Chaptr Homwork: (do at last 5 out of th following problms 4..4, 4..6, 4.., 4.3.4, 4.3.5, 4.4., 4.4.4, 4.5.3, 4.6.3, 4.6.7, 4.6., 4.7.9, 4.7., 4.8.3, 4.8.7, 4.9., 4.9.3, 4.9., 4..3, 4..8, 4.., 4.., Joint Cumulativ Distri. Function For r.v. s X and Y, th joint CDF is F X,Y (,y = P[X, Y y] Proprtis of joint CDF F X,Y (,y F X,Y (, = F X (, F X,Y (,y = F Y (y F X,Y (,- = F X,Y (-, = F X,Y (, = F X,Y (,y F X,Y (,y if and y y G. Qu ENEE 34 Enginring Probability
2 Joint Probability Mass Function For two discrt r.v. s X and Y, th joint PMF is P X,Y (,y = P[X=, Y=y] Eampl 4.*: Flip twic a biasd coin, whr had coms out with.9. X: th numbr of hads; Y: th numbr of hads bfor th first tail. Draw th tr diagram. Find th joint PMF in function form, points in th X-Y plan, 3 matri form. G. Qu ENEE 34 Enginring Probability 3 Evnt Probability For two discrt r.v. s X and Y and any st B in th X-Y plan, vnt {(,y S X,y S y, (,y B} happns with P[B] = (,y B P X,Y (,y Eampls: G. Qu ENEE 34 Enginring Probability 4
3 Marginal Prob. Mass Function For two discrt r.v. s X and Y, thir PMFs P X ( and P Y (y ar also calld marginal PMFs: P X (= y Sy P X,Y (,y, P Y (y= S P X,Y (,y for any S X, vnt {X= }={(,y =,y S y } Eampl 4.3: P X,Y (,y X= X= X= P Y (y Y=..9. Y=.9.9 Y= G. Qu ENEE 34 Enginring Probability 5 P X (. Evnt Probability and Joint PDF For two continuous r.v. s X and Y and any st A in th X-Y plan, vnt {(,y S X,y S y, (,y A} happns with P[A] = A f X,Y (,yddy f X,Y (,y=( / yf X,Y (,y is th joint PDF. f X,Y (,y y X, Y (, y = f X, Y F ( u, v dvdu P[ <X, y <Y y ] = F X,Y (,y - F X,Y (,y - F X,Y (,y + F X,Y (,y G. Qu ENEE 34 Enginring Probability 6 3
4 Eampls Random variabls X and Y hav joint PDF f X,Y (,y = c if y and o.w. (E. 4.4* What is c? (E. 4.5 (E. 4.6* What is joint CDF? What is P[A] = P[ X+Y ]? G. Qu ENEE 34 Enginring Probability 7 Marginal Prob. Dnsity Function f For two continuous r.v. s X and Y with joint PDF f X,Y (,y, th marginal PDFs f X ( and f Y (y ar th PDFs of X and Y, and w hav X ( f X, Y (, y dy fy ( y = f X, Y (, y = d Eampl 4.7: Find th marginal PDFs for th following joint PDF of X and Y: f X, Y 5y / 4 (, y =, othrwis y G. Qu ENEE 34 Enginring Probability 8 4
5 Functions of Two Discrt R.V. s For two discrt r.v. s X and Y, th random variabl W=g(X,Y has PMF P W (w= g(,y=w P X,Y (,y Eampl (Problm 4.6.4: Joint PMF: P X,Y (,y =. for,y, and othrwis. W=min(X,Y P W (w=? G. Qu ENEE 34 Enginring Probability 9 Functions of Two Continuous R.V. s For two continuous r.v. s X and Y, th random variabl W=g(X,Y has CDF F W (w=p[w w] = g(,y w f X,Y (,yddy Eampl 4.9: Joint PDF: f X,Y (,y = /5 for 5, y 3 and othrwis. W=ma(X,Y f W (w=? G. Qu ENEE 34 Enginring Probability 5
6 Epctd Valus of Drivd R.V. s For two r.v. s X and Y, th drivd random variabl W=g(X,Y has pctd valu: E[W] = g(,yp X,Y (,y E[W] = g(,yf X,Y (,yddy E[X+Y] = E[X] +E[Y] Eampl: X and Y discrt X and Y continuous Joint PDF: f X,Y (,y = /5 for 5, y 3 and othrwis. W=ma(X,Y E[W] = ma(,yf X,Y (,yddy = 4/5 E[W] = wf W (wdw = 4/5 G. Qu ENEE 34 Enginring Probability Covarianc of Two R.V. s Covarianc: Cov[X,Y] = E[(X- X (Y- Y ] Also known as σ XY X and Y ar uncorrlatd if Cov[X,Y]=. Corrlation Cofficint: X,Y =Cov[X,Y]/σ X σ Y Whn Y=X, Cov[X,Y]=Var[X], X,Y =. If Y=aX+b, X,Y =- if a<; if a=; if a>. Thorm 4.7: - X,Y G. Qu ENEE 34 Enginring Probability 6
7 Covarianc and Corrlation Corrlation: r X,Y =E[XY] X and Y ar orthogonal if r X,Y =. Whn Y=X, r X,Y =E[X ], th nd momnt Proprtis: Cov[X,Y] = r X,Y - X Y If X and Y ar orthogonal, Cov[X,Y]=- X Y Var[X+Y]=Var[X]+Var[Y]+Cov[X,Y] G. Qu ENEE 34 Enginring Probability 3 Conditioning on an Evnt: Discrt PMF: P X ( = P[X=] Conditional PMF: P X B ( = P[X= B] P X B ( = P X (/P[B] for B; othrwis. E[X B]= P X B ( E[W B]= g( P X B ( for W=g(X Joint PMF: P X,Y (,y = P[X=,Y=y] Conditional joint PMF: P X,Y B (,y = P[(X=,Y=y B] P X,Y B (,y = P X,Y (,y/p[b] for (,y B; othrwis. E[W B]= g(,y P X,Y B (,y for W=g(X,Y G. Qu ENEE 34 Enginring Probability 4 7
8 Conditioning on Evnt {Y=y} Joint PMF: P X,Y (,y = P[X=,Y=y] Conditional joint PMF: P X,Y B (,y = P[(X=,Y=y B] P X,Y B (,y = P X,Y (,y/p[b] for (,y B; othrwis. E[W B]= g(,y P X,Y B (,y for W=g(X,Y Whn B={Y=y} P[(X=,Y=y B] = P[X= B] = P[X= Y=y] P X Y ( y P X,Y (,y = P X Y ( yp Y (y = P Y X (y P X ( P[X= Y=y]=P[X=,Y=y]/P[Y=y] E[g(X,Y Y=y]= g(,y P X Y ( y G. Qu ENEE 34 Enginring Probability 5 Eampls R.V. s X and Y hav joint PMF P X,Y (,y=.5/ for Y X 4; and othrwis. B: X+Y 4. W=X+Y. (E. 4.3 find P X,Y B (,y (E. 4.5 find Var[W B] (E. 4.7 find P Y X (y (E. 4.8 find E[Y X=] G. Qu ENEE 34 Enginring Probability 6 8
9 Conditioning on an Evnt: Continuous CDF: F X ( = P[X ] PDF: f X ( = (d/d F X ( Conditional PDF: f X B ( = f X (/P[B] for B; othrwis. E[X B] = f X B (d E[W B] = g( f X B (d for W=g(X Joint PDF: f X,Y (,y =( / y F X,Y (,y Conditional joint PDF: f X,Y B (,y = f X,Y (,y/p[b] for (,y B with P[B]>; othrwis. E[W B]= g(,y f X,Y B (,yddy for W=g(X,Y G. Qu ENEE 34 Enginring Probability 7 Conditioning on Evnt {Y=y} Joint PDF: f X,Y (,y =( / y F X,Y (,y Conditional joint PDF: f X,Y B (,y = f X,Y (,y/p[b] for (,y B with P[B]>; othrwis. E[W B]= g(,y f X,Y B (,yddy Whn B={Y=y} for W=g(X,Y P[B] = P[Y=y] =, so cannot us th abov formula f X Y ( y f X,Y (,y/f Y (y, f X Y ( y f X,Y (,y/f X ( for f Y (y > and f X ( >. E[g(X,Y Y=y]= g(,y f X Y ( yd G. Qu ENEE 34 Enginring Probability 8 9
10 Eampl: Problm 4.9. Random variabls X and Y hav joint PDF f X,Y (,y =.5 if - y and o.w. What is marginal PDF f Y (y? f Y (y = f X,Y (,yd = (y+/ What is conditional PDF f X Y ( y? f X Y ( y=f X,Y (,y/f Y (y=/(y+ What is E[X Y=y]? E[X Y=y]= f X Y ( yd = (y-/ if - y if - y G. Qu ENEE 34 Enginring Probability 9 Conditional Varianc and Eampl For drivd r.v. W=g(X,Y and vnt B with P[B] >, Var[W B] = E[(W- W B B] = E[W B]- ( W B Eampls: Joint PDF: f X,Y (,y = /5 for 5, y 3 and othrwis. W=XY. B: X+Y 4. (E. 4.4 find f X,Y B (,y (E. 4.6 find E[W B] and Var[W B] G. Qu ENEE 34 Enginring Probability
11 Itratd Epctation E[X Y=y] = f X Y ( yd is a function of Y, dnot it as E[X Y]. In Problm 4.9., E[X Y](y = E[X Y=y] = (y-/ Itratd pctation: E[E[X Y]]=E[X] Proof: E[E[X Y]] = E[X Y=y] f Y (y dy = ( f X Y ( yd f Y (ydy = f X Y ( y f Y (y d dy = ( f X Y ( y f Y (y dy d = f X (d In gnral, E[E[g(X Y]]=E[g(X] G. Qu ENEE 34 Enginring Probability Indpndnt Random Variabls Rcall: two vnts ar indpndnt iff P[AB] = P[A]P[B]. Also P[A B]=P[A] Two r.v. s X and Y ar indpndnt iff P X,Y (,y = P X (P Y (y f X,Y (,y = f X (f Y (y discrt Whn X and Y ar indpndnt, continuous P X Y ( y= P X,Y (,y/p Y (y=p X (, P Y X (y = P Y (y f X Y ( y= f X,Y (,y/f Y (y=f X (, f Y X (y = f Y (y G. Qu ENEE 34 Enginring Probability
12 Proprtis of Indpndnt R.V. s E[g(Xh(Y] = E[g(X]E[h(Y] r X,Y =E[XY]=E[X]E[Y] Cov[X.Y] = X,Y = indpndnt uncorrlatd but not vic vrsa (s. 4.5 Var[X+Y] = Var[] + Var[Y] E[X Y=y] = E[X] for all y S Y E[Y X=] = E[Y] for all X S G. Qu ENEE 34 Enginring Probability 3 Quiz 4. R.V. s X and Y hav th following joint PMF, ar thy indpndnt? P X,Y (,y X= X= X= Y=..9 Y=.9 Y=.8 P X (..8.8 P Y (y..9.8 R.V. s X and Y ar indpndnt and idntically distributd with PDF as follows. What is th joint PDF? f X ( = What is th CDF of Z=ma(X,Y? othrwis G. Qu ENEE 34 Enginring Probability 4
13 3 G. Qu ENEE 34 Enginring Probability 5 Bivariat Gaussian R.V. s Rcall Gaussian (,σ X: E[X]=, Var[X]=σ Bivariat Gaussian Random Variabls X and Y hav PDF f X,Y (,y with 5 paramtrs,, σ >, σ > and -<< dfind as: ( ( σ πσ = X f ( ( ( ( ( ( ( σ σ σ πσ σ πσ y y G. Qu ENEE 34 Enginring Probability 6 Bivariat Gaussian PDF: impact of = : circular symmtric < : ridg ovr lin =-y > : ridg ovr lin =y : masurs th pak of th ridg at (=, y= ( ( ( ( ( ( ( σ σ σ πσ σ πσ y y
14 Bivariat Gaussian PDF is a PDF πσ ( σ ( f X,Y (,y f X,Y (,y= πσ ( y σ ( Dfin: ( = +σ (- /σ, σ= σ (- / ( ( y σ σ ( f X,Y (,y can b rwrittn as th product of PDFs of Gaussian(,σ and Gaussian((,σ: ( ( y ( σ σ πσ πσ G. Qu ENEE 34 Enginring Probability 7 R.V. s X and Y in Bivariat Gaussian For bivariat Gaussian R.V. s X and Y, X is Gaussian (,σ Y is Gaussian (,σ. X,Y = Y X is Gaussian (,σ. X Y is also Gaussian. X,Y = iff X and Y ar indpndnt. G. Qu ENEE 34 Enginring Probability 8 4
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