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

${f}_{XY}\left(t,u\right)=\frac{3}{88}\left(2t+3{u}^{2}\right)$ for $0\le t\le 2$ , $0\le u\le 1+t$ (see Exercise 15 from "Problems on Random Vectors and Joint Distributions").

$Z={I}_{\left[0,1\right]}\left(X\right)4X+{I}_{\left(1,2\right]}\left(X\right)\left(X+Y\right)$

Determine $P\left(Z\le 2\right)$

$P\left(Z\le 2\right)=P\left(Z\in Q=Q1M1\bigvee Q2M2\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{where}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M1=\left\{\left(t,u\right):0\le t\le 1,\phantom{\rule{0.277778em}{0ex}}0\le u\le 1+t\right\}$
$M2=\left\{\left(t,u\right):1
$Q1=\left\{\left(t,u\right):0\le t\le 1/2\right\},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}Q2=\left\{\left(t,u\right):u\le 2-t\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{(see}\phantom{\rule{4.pt}{0ex}}\text{figure)}$
$P=\frac{3}{88}{\int }_{0}^{1/2}{\int }_{0}^{1+t}\left(2t+3{u}^{2}\right)\phantom{\rule{0.166667em}{0ex}}dudt+\frac{3}{88}{\int }_{1}^{2}{\int }_{0}^{2-t}\left(2t+3{u}^{2}\right)\phantom{\rule{0.166667em}{0ex}}dudt=\frac{563}{5632}$
tuappr Enter matrix [a b]of X-range endpoints [0 2] Enter matrix [c d]of Y-range endpoints [0 3] Enter number of X approximation points 200Enter number of Y approximation points 300 Enter expression for joint density (3/88)*(2*t + 3*u.^2).*(u<=1+t) Use array operations on X, Y, PX, PY, t, u, and PG = 4*t.*(t<=1) + (t+u).*(t>1); [Z,PZ]= csort(G,P); PZ2 = (Z<=2)*PZ' PZ2 = 0.1010 % Theoretical = 563/5632 = 0.1000

${f}_{XY}\left(t,u\right)=\frac{24}{11}tu$ for $0\le t\le 2$ , $0\le u\le min\left\{1,2-t\right\}$ (see Exercise 17 from "Problems on Random Vectors and Joint Distributions").

$Z={I}_{M}\left(X,Y\right)\frac{1}{2}X+{I}_{{M}^{c}}\left(X,Y\right){Y}^{2},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M=\left\{\left(t,u\right):u>t\right\}$

Determine $P\left(Z\le 1/4\right)$ .

$P\left(Z\le 1/4\right)=P\left(\left(X,Y\right)\in {M}_{1}{Q}_{1}\bigvee {M}_{2}{Q}_{2}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{1}=\left\{\left(t,u\right):0\le t\le u\le 1\right\}$
${M}_{2}=\left\{\left(t,u\right):0\le t\le 2,\phantom{\rule{0.277778em}{0ex}}0\le t\le min\left(t,2-t\right)\right\}$
${Q}_{1}=\left\{\left(t,u\right):t\le 1/2\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{Q}_{2}=\left\{\left(t,u\right):u\le 1/2\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{(see}\phantom{\rule{4.pt}{0ex}}\text{figure)}$
$P=\frac{24}{11}{\int }_{0}^{1/2}{\int }_{0}^{1}tu\phantom{\rule{0.166667em}{0ex}}dudt+\frac{24}{11}{\int }_{1/2}^{3/2}{\int }_{0}^{1/2}tu\phantom{\rule{0.166667em}{0ex}}dudt+\frac{24}{11}{\int }_{3/2}^{2}{\int }_{0}^{2-t}tu\phantom{\rule{0.166667em}{0ex}}dudt=\frac{85}{176}$
tuappr Enter matrix [a b]of X-range endpoints [0 2] Enter matrix [c d]of Y-range endpoints [0 1] Enter number of X approximation points 400Enter number of Y approximation points 200 Enter expression for joint density (24/11)*t.*u.*(u<=min(1,2-t)) Use array operations on X, Y, PX, PY, t, u, and PG = 0.5*t.*(u>t) + u.^2.*(u<t); [Z,PZ]= csort(G,P); pp = (Z<=1/4)*PZ' pp = 0.4844 % Theoretical = 85/176 = 0.4830

${f}_{XY}\left(t,u\right)=\frac{3}{23}\left(t+2u\right)$ for $0\le t\le 2$ , $0\le u\le max\left\{2-t,t\right\}$ (see Exercise 18 from "Problems on Random Vectors and Joint Distributions").

$Z={I}_{M}\left(X,Y\right)\left(X+Y\right)+{I}_{{M}^{c}}\left(X,Y\right)2Y,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M=\left\{\left(t,u\right):max\left(t,u\right)\le 1\right\}$

Determine $P\left(Z\le 1\right)$ .

$P\left(Z\le 1\right)=P\left(\left(X,Y\right)\in {M}_{1}{Q}_{1}\bigvee {M}_{2}{Q}_{2}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{1}=\left\{\left(t,u\right):0\le t\le 1,\phantom{\rule{0.277778em}{0ex}}0\le u\le 1-t\right\}$
${M}_{2}=\left\{\left(t,u\right):1\le t\le 2,\phantom{\rule{0.277778em}{0ex}}0\le u\le t\right\}$
${Q}_{1}=\left\{\left(t,u\right):u\le 1-t\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{Q}_{2}=\left\{\left(t,u\right):u\le 1/2\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{(see}\phantom{\rule{4.pt}{0ex}}\text{figure)}$
$P=\frac{3}{23}{\int }_{0}^{1}{\int }_{0}^{1-t}\left(t+2u\right)\phantom{\rule{0.166667em}{0ex}}dudt+\frac{3}{23}{\int }_{1}^{2}{\int }_{0}^{1/2}\left(t+2u\right)\phantom{\rule{0.166667em}{0ex}}dudt=\frac{9}{46}$
tuappr Enter matrix [a b]of X-range endpoints [0 2] Enter matrix [c d]of Y-range endpoints [0 2] Enter number of X approximation points 300Enter number of Y approximation points 300 Enter expression for joint density (3/23)*(t + 2*u).*(u<=max(2-t,t)) Use array operations on X, Y, PX, PY, t, u, and PM = max(t,u)<= 1; G = M.*(t + u) + (1 - M)*2.*u;p = total((G<=1).*P) p = 0.1960 % Theoretical = 9/46 = 0.1957

${f}_{XY}\left(t,u\right)=\frac{12}{179}\left(3{t}^{2}+u\right)$ , for $0\le t\le 2$ , $0\le u\le min\left\{2,3-t\right\}$ (see Exercise 19 from "Problems on Random Vectors and Joint Distributions").

$Z={I}_{M}\left(X,Y\right)\left(X+Y\right)+{I}_{{M}^{c}}\left(X,Y\right)2{Y}^{2},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M=\left\{\left(t,u\right):t\le 1,\phantom{\rule{0.277778em}{0ex}}u\ge 1\right\}$

Determine $P\left(Z\le 2\right)$ .

$P\left(Z\le 2\right)=P\left(\left(,Y\right)\in {M}_{1}{Q}_{1}\bigvee \left({M}_{2}\bigvee {M}_{3}\right){Q}_{2}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{1}=\left\{\left(t,u\right):0\le t\le 1,\phantom{\rule{0.277778em}{0ex}}1\le u\le 2\right\}$
${M}_{2}=\left\{\left(t,u\right):0\le t\le 1,\phantom{\rule{0.277778em}{0ex}}0\le u\le 1\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{3}=\left\{\left(t,u\right):1\le t\le 2,\phantom{\rule{0.277778em}{0ex}}0\le u\le 3-t\right\}$
${Q}_{1}=\left\{\left(t,u\right):u\le 1-t\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{Q}_{2}=\left\{\left(t,u\right):u\le 1/2\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{(see}\phantom{\rule{4.pt}{0ex}}\text{figure)}$
$P=\frac{12}{179}{\int }_{0}^{1}{\int }_{0}^{2-t}\left(3{t}^{2}+u\right)\phantom{\rule{0.166667em}{0ex}}dudt+\frac{12}{179}{\int }_{1}^{2}{\int }_{0}^{1}\left(3{t}^{2}+u\right)\phantom{\rule{0.166667em}{0ex}}dudt=\frac{119}{179}$
tuappr Enter matrix [a b]of X-range endpoints [0 2] Enter matrix [c d]of Y-range endpoints [0 2] Enter number of X approximation points 300Enter number of Y approximation points 300 Enter expression for joint density (12/179)*(3*t.^2 + u).*(u<=min(2,3-t)) Use array operations on X, Y, PX, PY, t, u, and PM = (t<=1)&(u>=1); Z = M.*(t + u) + (1 - M)*2.*u.^2;G = M.*(t + u) + (1 - M)*2.*u.^2; p = total((G<=2).*P) p = 0.6662 % Theoretical = 119/179 = 0.6648

${f}_{XY}\left(t,u\right)=\frac{12}{227}\left(3t+2tu\right)$ , for $0\le t\le 2$ , $0\le u\le min\left\{1+t,2\right\}$ (see Exercise 20 from "Problems on Random Variables and Joint Distributions")

$Z={I}_{M}\left(X,Y\right)X+{I}_{{M}^{c}}\left(X,Y\right)\frac{Y}{X},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M=\left\{\left(t,u\right):u\le min\left(1,2-t\right)\right\}$

Detemine $P\left(Z\le 1\right)$ .

$P\left(Z\le 1\right)=P\left(\left(X,Y\right)\in {M}_{1}{Q}_{1}\bigvee {M}_{2}{Q}_{2}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{1}=M,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{2}={M}^{c}$
${Q}_{1}=\left\{\left(t,u\right):0\le t\le 1\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{Q}_{2}=\left\{\left(t,u\right):u\le t\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{(see}\phantom{\rule{4.pt}{0ex}}\text{figure)}$
$P=\frac{12}{227}{\int }_{0}^{1}{\int }_{0}^{1}\left(3t+2tu\right)\phantom{\rule{0.166667em}{0ex}}dudt+\frac{12}{227}{\int }_{1}^{2}{\int }_{2-t}^{t}\left(3t+2tu\right)\phantom{\rule{0.166667em}{0ex}}dudt=\frac{124}{227}$
tuappr Enter matrix [a b]of X-range endpoints [0 2] Enter matrix [c d]of Y-range endpoints [0 2] Enter number of X approximation points 400Enter number of Y approximation points 400 Enter expression for joint density (12/227)*(3*t+2*t.*u).*(u<=min(1+t,2)) Use array operations on X, Y, PX, PY, t, u, and PQ = (u<=1).*(t<=1) + (t>1).*(u>=2-t).*(u<=t); P = total(Q.*P)P = 0.5478 % Theoretical = 124/227 = 0.5463

The class $\left\{X,\phantom{\rule{0.166667em}{0ex}}Y,\phantom{\rule{0.166667em}{0ex}}Z\right\}$ is independent.

$X=-2{I}_{A}+{I}_{B}+3{I}_{C}$ . Minterm probabilities are (in the usual order)

$0.255\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0.025\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0.375\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0.045\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0.108\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0.012\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0.162\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0.018$

$Y={I}_{D}+3{I}_{E}+{I}_{F}-3$ . The class $\left\{D,\phantom{\rule{0.166667em}{0ex}}E,\phantom{\rule{0.166667em}{0ex}}F\right\}$ is independent with

$P\left(D\right)=0.32\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left(E\right)=0.56\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left(F\right)=0.40$

Z has distribution

 Value -1.3 1.2 2.7 3.4 5.8 Probability 0.12 0.24 0.43 0.13 0.08

Determine $P\left({X}^{2}+3X{Y}^{2}>3Z\right)$ .

% file npr10_16.m Data for [link] cx = [-2 1 3 0];pmx = 0.001*[255 25 375 45 108 12 162 18];cy = [1 3 1 -3];pmy = minprob(0.01*[32 56 40]);Z = [-1.3 1.2 2.7 3.4 5.8];PZ = 0.01*[12 24 43 13 8];disp('Data are in cx, pmx, cy, pmy, Z, PZ') npr10_16 % Call for dataData are in cx, pmx, cy, pmy, Z, PZ [X,PX]= canonicf(cx,pmx); [Y,PY]= canonicf(cy,pmy); icalc3Enter row matrix of X-values X Enter row matrix of Y-values YEnter row matrix of Z-values Z Enter X probabilities PXEnter Y probabilities PY Enter Z probabilities PZUse array operations on matrices X, Y, Z, PX, PY, PZ, t, u, v, and PM = t.^2 + 3*t.*u.^2>3*v; PM = total(M.*P)PM = 0.3587

The simple random variable X has distribution

$X=\left[-3.1\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}-0.5\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}1.2\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}2.4\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}3.7\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}4.9\right]\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}PX=\left[0.15\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0.22\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0.33\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0.12\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0.11\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0.07\right]$
1. Plot the distribution function F X and the quantile function Q X .
2. Take a random sample of size $n=10,000$ . Compare the relative frequency for each value with the probability that value is taken on.
X = [-3.1 -0.5 1.2 2.4 3.7 4.9];PX = 0.01*[15 22 33 12 11 7];ddbn Enter row matrix of VALUES XEnter row matrix of PROBABILITIES PX % Plot not reproduced here dquanplotEnter VALUES for X X Enter PROBABILITIES for X PX % Plot not reproduced hererand('seed',0) % Reset random number generator dsample % for comparison purposesEnter row matrix of VALUES X Enter row matrix of PROBABILITIES PXSample size n 10000 Value Prob Rel freq-3.1000 0.1500 0.1490 -0.5000 0.2200 0.21641.2000 0.3300 0.3340 2.4000 0.1200 0.11843.7000 0.1100 0.1070 4.9000 0.0700 0.0752Sample average ex = 0.8792 Population mean E[X]= 0.859 Sample variance vx = 5.146Population variance Var[X] = 5.112

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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive