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(See Exercise 6 from "Problems on Random Vectors and Joint Distributions", and Exercise 1 from "Problems on Independent Classes of Random Variables")) The pair { X , Y } has the joint distribution

(in m-file npr08_06.m ):

X = [ - 2 . 3 - 0 . 7 1 . 1 3 . 9 5 . 1 ] Y = [ 1 . 3 2 . 5 4 . 1 5 . 3 ]
P = 0 . 0483 0 . 0357 0 . 0420 0 . 0399 0 . 0441 0 . 0437 0 . 0323 0 . 0380 0 . 0361 0 . 0399 0 . 0713 0 . 0527 0 . 0620 0 . 0609 0 . 0551 0 . 0667 0 . 0493 0 . 0580 0 . 0651 0 . 0589

Determine P ( max { X , Y } 4 ) , P ( | X - Y | > 3 ) . Let Z = 3 X 3 + 3 X 2 Y - Y 3 .
Determine P ( Z < 0 ) and P ( - 5 < Z 300 ) .

npr08_06 Data are in X, Y, P jcalcEnter JOINT PROBABILITIES (as on the plane) P Enter row matrix of VALUES of X XEnter row matrix of VALUES of Y Y Use array operations on matrices X, Y, PX, PY, t, u, and PP1 = total((max(t,u)<=4).*P) P1 = 0.4860P2 = total((abs(t-u)>3).*P) P2 = 0.4516G = 3*t.^3 + 3*t.^2.*u - u.^3; P3 = total((G<0).*P) P3 = 0.5420P4 = total(((-5<G)&(G<=300)).*P) P4 = 0.3713[Z,PZ] = csort(G,P); % Alternate: use dbn for Zp4 = ((-5<Z)&(Z<=300))*PZ' p4 = 0.3713
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(See Exercise 2 from "Problems on Independent Classes of Random Variables") The pair { X , Y } has the joint distribution (in m-file npr09_02.m ):

X = [ - 3 . 9 - 1 . 7 1 . 5 2 8 4 . 1 ] Y = [ - 2 1 2 . 6 5 . 1 ]
P = 0 . 0589 0 . 0342 0 . 0304 0 . 0456 0 . 0209 0 . 0961 0 . 0556 0 . 0498 0 . 0744 0 . 0341 0 . 0682 0 . 0398 0 . 0350 0 . 0528 0 . 0242 0 . 0868 0 . 0504 0 . 0448 0 . 0672 0 . 0308

Determine P ( { X + Y 5 } { Y 2 } ) , P ( X 2 + Y 2 10 ) .

npr09_02 Data are in X, Y, P jcalcEnter JOINT PROBABILITIES (as on the plane) P Enter row matrix of VALUES of X XEnter row matrix of VALUES of Y Y Use array operations on matrices X, Y, PX, PY, t, u, and PM1 = (t+u>=5)|(u<=2); P1 = total(M1.*P)P1 = 0.7054 M2 = t.^2 + u.^2<= 10; P2 = total(M2.*P)P2 = 0.3282
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(See Exercise 7 from "Problems on Random Vectors and Joint Distributions", and Exercise 3 from "Problems on Independent Classes of Random Variables") The pair { X , Y } has the joint distribution

(in m-file npr08_07.m ):

P ( X = t , Y = u )
t = -3.1 -0.5 1.2 2.4 3.7 4.9
u = 7.5 0.0090 0.0396 0.0594 0.0216 0.0440 0.0203
4.1 0.0495 0 0.1089 0.0528 0.0363 0.0231
-2.0 0.0405 0.1320 0.0891 0.0324 0.0297 0.0189
-3.8 0.0510 0.0484 0.0726 0.0132 0 0.0077

Determine P ( X 2 - 3 X 0 ) , P ( X 3 - 3 | Y | < 3 ) .

npr08_07 Data are in X, Y, P jcalcEnter JOINT PROBABILITIES (as on the plane) P Enter row matrix of VALUES of X XEnter row matrix of VALUES of Y Y Use array operations on matrices X, Y, PX, PY, t, u, and PM1 = t.^2 - 3*t<=0; P1 = total(M1.*P)P1 = 0.4500 M2 = t.^3 - 3*abs(u)<3; P2 = total(M2.*P)P2 = 0.7876
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For the pair { X , Y } in [link] , let Z = g ( X , Y ) = 3 X 2 + 2 X Y - Y 2 . Determine and plot the distribution function for Z .

G = 3*t.^2 + 2*t.*u - u.^2; % Determine g(X,Y) [Z,PZ]= csort(G,P); % Obtain dbn for Z = g(X,Y) ddbn % Call for plotting m-procedureEnter row matrix of VALUES Z Enter row matrix of PROBABILITIES PZ % Plot not reproduced here
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For the pair { X , Y } in [link] , let

W = g ( X , Y ) = X for X + Y 4 2 Y for X + Y > 4 = I M ( X , Y ) X + I M c ( X , Y ) 2 Y

Determine and plot the distribution function for W .

H = t.*(t+u<=4) + 2*u.*(t+u>4); [W,PW]= csort(H,P); ddbnEnter row matrix of VALUES W Enter row matrix of PROBABILITIES PW % Plot not reproduced here
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For the distributions in Exercises 10-15 below

  1. Determine analytically the indicated probabilities.
  2. Use a discrete approximation to calculate the same probablities.'

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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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