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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 has the joint distribution
(in m-file npr08_06.m ):
Determine
. Let
.
Determine
and
.
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 (See Exercise 2 from "Problems on Independent Classes of Random Variables") The pair has the joint distribution (in m-file npr09_02.m ):
Determine , .
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 (See Exercise 7 from "Problems on Random Vectors and Joint Distributions", and Exercise 3 from "Problems on Independent Classes of Random Variables") The pair has the joint distribution
(in m-file npr08_07.m ):
| 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 , .
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 For the pair in [link] , let . 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 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 For the distributions in Exercises 10-15 below
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