# 10.3 Problems on functions of random variables  (Page 2/3)

(See Exercise 6 from "Problems on Random Vectors and Joint Distributions", and Exercise 1 from "Problems on Independent Classes of Random Variables")) The pair $\left\{X,\phantom{\rule{0.166667em}{0ex}}Y\right\}$ has the joint distribution

(in m-file npr08_06.m ):

$X=\left[-2.3\phantom{\rule{0.277778em}{0ex}}-0.7\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}1.1\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}3.9\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}5.1\right]\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}Y=\left[1.3\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}2.5\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}4.1\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}5.3\right]$
$P=\left[\begin{array}{ccccc}0.0483\hfill & 0.0357\hfill & 0.0420\hfill & 0.0399\hfill & 0.0441\hfill \\ 0.0437\hfill & 0.0323\hfill & 0.0380\hfill & 0.0361\hfill & 0.0399\hfill \\ 0.0713\hfill & 0.0527\hfill & 0.0620\hfill & 0.0609\hfill & 0.0551\hfill \\ 0.0667\hfill & 0.0493\hfill & 0.0580\hfill & 0.0651\hfill & 0.0589\hfill \end{array}\right]$

Determine $P\left(max\left\{X,Y\right\}\le 4\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left(|X-Y|>3\right)$ . Let $Z=3{X}^{3}+3{X}^{2}Y-{Y}^{3}$ .
Determine $P\left(Z<0\right)$ and $P\left(-5 .

 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 $\left\{X,\phantom{\rule{0.166667em}{0ex}}Y\right\}$ has the joint distribution (in m-file npr09_02.m ):

$X=\left[-3.9\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}-1.7\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}1.5\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}2\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}8\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}4.1\right]\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}Y=\left[-2\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}1\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}2.6\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}5.1\right]$
$P=\left[\begin{array}{ccccc}0.0589\hfill & 0.0342\hfill & 0.0304\hfill & 0.0456\hfill & 0.0209\hfill \\ 0.0961\hfill & 0.0556\hfill & 0.0498\hfill & 0.0744\hfill & 0.0341\hfill \\ 0.0682\hfill & 0.0398\hfill & 0.0350\hfill & 0.0528\hfill & 0.0242\hfill \\ 0.0868\hfill & 0.0504\hfill & 0.0448\hfill & 0.0672\hfill & 0.0308\hfill \end{array}\right]$

Determine $P\left(\left\{X+Y\ge 5\right\}\cup \left\{Y\le 2\right\}\right)$ , $P\left({X}^{2}+{Y}^{2}\le 10\right)$ .

 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 $\left\{X,\phantom{\rule{0.166667em}{0ex}}Y\right\}$ has the joint distribution

(in m-file npr08_07.m ):

$P\left(X=t,\phantom{\rule{0.277778em}{0ex}}Y=u\right)$
 t = -3.1 -0.5 1.2 2.4 3.7 4.9 u = 7.5 0.009 0.0396 0.0594 0.0216 0.044 0.0203 4.1 0.0495 0 0.1089 0.0528 0.0363 0.0231 -2.0 0.0405 0.132 0.0891 0.0324 0.0297 0.0189 -3.8 0.051 0.0484 0.0726 0.0132 0 0.0077

Determine $P\left({X}^{2}-3X\le 0\right)$ , $P\left({X}^{3}-3|Y|<3\right)$ .

 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 $\left\{X,\phantom{\rule{0.166667em}{0ex}}Y\right\}$ in [link] , let $Z=g\left(X,Y\right)=3{X}^{2}+2XY-{Y}^{2}$ . 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

For the pair $\left\{X,\phantom{\rule{0.166667em}{0ex}}Y\right\}$ in [link] , let

$W=g\left(X,Y\right)=\left\{\begin{array}{cc}X& \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}X+Y\le 4\hfill \\ 2Y& \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}X+Y>4\hfill \end{array}\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}={I}_{M}\left(X,Y\right)X+{I}_{{M}^{c}}\left(X,Y\right)2Y$

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

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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