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8.3: Problemas en Vectores Aleatorios y Distribuciones Conjuntas

  • Page ID
    150929
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Ejercicio\(\PageIndex{1}\)

    Se seleccionan dos cartas al azar, sin reemplazo, de una baraja estándar. \(X\)Sea el número de ases y\(Y\) sea el número de espadas. Bajo los supuestos habituales, determinar la distribución conjunta y los marginales.

    Contestar

    \(X\)Sea el número de ases y\(Y\) sea el número de espadas. Definir los eventos\(AS_i\)\(A_i\),\(S_i\),, y\(N_i\),\(i = 1, 2\) de dibujar as de espadas, otro as, pala (que no sea el as), y ninguno en la selección i. Vamos\(P(i, k) = P(X = i, Y = k)\).

    \(P(0, 0) = P(N_1N_2) = \dfrac{36}{52} \cdot \dfrac{35}{51} = \dfrac{1260}{2652}\)

    \(P(0, 1) = P(N_1S_2 \bigvee S_1N_2) = \dfrac{36}{52} \cdot \dfrac{12}{51} + \dfrac{12}{52} \cdot \dfrac{36}{51} = \dfrac{864}{2652}\)

    \(P(0, 2) = P(S_1 S_2) = \dfrac{12}{52} \cdot \dfrac{11}{51} = \dfrac{132}{2652}\)

    \(P(1, 0) = P(A_N_2 \bigvee N_1 S_2) = \dfrac{3}{52} \cdot \dfrac{36}{51} + \dfrac{36}{52} \cdot \dfrac{3}{51} = \dfrac{216}{2652}\)

    \(P(1, 1) = P(A_1S_2 \bigvee S_1A_2 \bigvee AS_1N_2 \bigvee N_1AS_2) = \dfrac{3}{52} \cdot \dfrac{12}{51} + \dfrac{12}{52} \cdot \dfrac{3}{51} + \dfrac{1}{52} \cdot \dfrac{36}{51} + \dfrac{36}{52} \cdot \dfrac{1}{51} = \dfrac{144}{2652}\)

    \(P(1, 2) = P(AS_1S_2 \bigvee S_1AS_2) = \dfrac{1}{52} \cdot \dfrac{12}{51} + \dfrac{12}{52} \cdot \dfrac{1}{51} = \dfrac{24}{2652}\)

    \(P(2, 0) = P(A_1A_2) = \dfrac{3}{52} \cdot \dfrac{2}{51} = \dfrac{6}{2652}\)

    \(P(2, 1) = P(AS_1A_2 \bigvee A_1AS_2) = \dfrac{1}{52} \cdot \dfrac{3}{51} + \dfrac{3}{52} \cdot \dfrac{1}{51} = \dfrac{6}{2652}\)

    \(P(2, 2) = P(\emptyset) = 0\)

    % type npr08_01
    % file npr08_01.m
    % Solution for Exercise 8.3.1.
    X = 0:2;
    Y = 0:2;
    Pn = [132  24   0; 864 144  6; 1260 216 6];
    P = Pn/(52*51);
    disp('Data in Pn, P, X, Y')
     
    npr08_01         % Call for mfile
    Data in Pn, P, X, Y    % Result
    PX = sum(P)
    PX =  0.8507    0.1448    0.0045
    PY = fliplr(sum(P'))
    PY =  0.5588    0.3824    0.0588

    Ejercicio\(\PageIndex{2}\)

    Dos puestos para trabajos de campus están abiertos. Aplican dos estudiantes de segundo año, tres juniors y tres seniors. Se decide seleccionar dos al azar (cada par posible igualmente probable). \(X\)Sea el número de alumnos de segundo año y\(Y\) sea el número de juniors que sean seleccionados. Determinar la distribución conjunta para el par\(\{X, Y\}\) y a partir de esto determinar los marginales para cada uno.

    Contestar

    Dejen\(A_i, B_i, C_i\) ser los eventos de seleccionar a un segundo, junior, o senior, respectivamente, en el\(i\) th juicio. Dejar\(X\) ser el número de estudiantes de segundo año y\(Y\) ser el número de juniors seleccionados.

    Set\(P(i, k) = P(X = i, Y = k)\)

    \(P(0, 0) = P(C_1C_2) = \dfrac{3}{8} \cdot \dfrac{2}{7} = \dfrac{6}{56}\)

    \(P(0, 1) = P(B_1C_2) + P(C_1B_2) = \dfrac{3}{8} \cdot \dfrac{3}{7} + \dfrac{3}{8} \cdot \dfrac{3}{7} = \dfrac{18}{56}\)

    \(P(0, 2) = P(B_1B_2) = \dfrac{3}{8} \cdot \dfrac{2}{7} = \dfrac{6}{56}\)

    \(P(1, 0) = P(A_1C_2) + P(C_1A_2) = \dfrac{2}{8} \cdot \dfrac{3}{7} + \dfrac{3}{8} \cdot \dfrac{2}{7} = \dfrac{12}{56}\)

    \(P(1, 1) = P(A_1B_2) + P(B_1A_2) = \dfrac{2}{8} \cdot \dfrac{3}{7} + \dfrac{3}{8} \cdot \dfrac{2}{7} = \dfrac{12}{56}\)

    \(P(2, 0) = P(A_1A_2) = \dfrac{2}{8} \cdot \dfrac{1}{7} = \dfrac{2}{56}\)

    \(P(1, 2) = P(2, 1) = P(2, 2) = 0\)

    \(PX =\)[30/56 24/56 2/56]\(PY =\) [20/56 30/56 6/56]

    % file npr08_02.m
    % Solution for Exercise 8.3.2.
    X = 0:2;
    Y = 0:2;
    Pn = [6 0 0; 18 12 0; 6 12 2];
    P = Pn/56;
    disp('Data are in X, Y,Pn, P')
    npr08_02
    Data are in X, Y,Pn, P
    PX = sum(P)
    PX =  0.5357    0.4286    0.0357
    PY = fliplr(sum(P'))
    PY =  0.3571    0.5357    0.1071

    Ejercicio\(\PageIndex{3}\)

    Se enrolla un dado. Deja\(X\) ser el número que aparece. Una moneda es volteada\(X\) veces. \(Y\)Sea el número de cabezas que aparecen. Determinar la distribución conjunta para el par\(\{X, Y\}\). Asumir\(P(X = k) = 1/6\) para\(1 \le k \le 6\) y para cada uno\(k\),\(P(Y = j|X = k)\) tiene la distribución binomial (\(k\), 1/2). Organizar la matriz conjunta como en el plano, con valores de\(Y\) aumento hacia arriba. Determinar la distribución marginal para\(Y\). (Para una forma basada en MATLAB de determinar la distribución conjunta, consulte el Ejemplo 14.1.7 de “Expectativa condicional, regresión”)

    Contestar

    \(P(X = i, Y = k) = P(X = i) P(Y = k|X = i) = (1/6) P(Y = k|X = i)\).

    % file npr08_03.m
    % Solution for Exercise 8.3.3.
    X = 1:6;
    Y = 0:6;
    P0 = zeros(6,7);       % Initialize
    for i = 1:6            % Calculate rows of Y probabilities
        P0(i,1:i+1) = (1/6)*ibinom(i,1/2,0:i);
    end
    P = rot90(P0);         % Rotate to orient as on the plane
    PY = fliplr(sum(P'));  % Reverse to put in normal order
    disp('Answers are in X, Y, P, PY')
    npr08_03            % Call for solution m-file
    Answers are in X, Y, P, PY
    disp(P)
             0         0         0         0         0    0.0026
             0         0         0         0    0.0052    0.0156
             0         0         0    0.0104    0.0260    0.0391
             0         0    0.0208    0.0417    0.0521    0.0521
             0    0.0417    0.0625    0.0625    0.0521    0.0391
        0.0833    0.0833    0.0625    0.0417    0.0260    0.0156
        0.0833    0.0417    0.0208    0.0104    0.0052    0.0026
    disp(PY)
         0.1641  0.3125  0.2578  0.1667  0.0755  0.0208  0.0026

    Ejercicio\(\PageIndex{4}\)

    Como variación del Ejercicio 8.3.3. , Supongamos que se tira un par de dados en lugar de un solo dado. Determinar la distribución conjunta para el par\(\{X, Y\}\) y a partir de esto determinar la distribución marginal para\(Y\).

    Contestar
    % file npr08_04.m
    % Solution for Exercise 8.3.4.
    X = 2:12;
    Y = 0:12;
    PX = (1/36)*[1 2 3 4 5 6 5 4 3 2 1];
    P0 = zeros(11,13);
    for i = 1:11
        P0(i,1:i+2) = PX(i)*ibinom(i+1,1/2,0:i+1);
    end
    P = rot90(P0);
    PY = fliplr(sum(P'));
    disp('Answers are in X, Y, PY, P')
    npr08_04
    Answers are in X, Y, PY, P
    disp(P)
      Columns 1 through 7
             0         0         0         0         0         0         0
             0         0         0         0         0         0         0
             0         0         0         0         0         0         0
             0         0         0         0         0         0         0
             0         0         0         0         0         0    0.0005
             0         0         0         0         0    0.0013    0.0043
             0         0         0         0    0.0022    0.0091    0.0152
             0         0         0    0.0035    0.0130    0.0273    0.0304
             0         0    0.0052    0.0174    0.0326    0.0456    0.0380
             0    0.0069    0.0208    0.0347    0.0434    0.0456    0.0304
        0.0069    0.0208    0.0312    0.0347    0.0326    0.0273    0.0152
        0.0139    0.0208    0.0208    0.0174    0.0130    0.0091    0.0043
        0.0069    0.0069    0.0052    0.0035    0.0022    0.0013    0.0005
      Columns 8 through 11
             0         0         0    0.0000
             0         0    0.0000    0.0001
             0    0.0001    0.0003    0.0004
        0.0002    0.0008    0.0015    0.0015
        0.0020    0.0037    0.0045    0.0034
        0.0078    0.0098    0.0090    0.0054
        0.0182    0.0171    0.0125    0.0063
        0.0273    0.0205    0.0125    0.0054
        0.0273    0.0171    0.0090    0.0034
        0.0182    0.0098    0.0045    0.0015
        0.0078    0.0037    0.0015    0.0004
        0.0020    0.0008    0.0003    0.0001
        0.0002    0.0001    0.0000    0.0000
    disp(PY)
      Columns 1 through 7
        0.0269    0.1025    0.1823    0.2158    0.1954    0.1400    0.0806
      Columns 8 through 13
        0.0375    0.0140    0.0040    0.0008    0.0001    0.0000

    Ejercicio\(\PageIndex{5}\)

    Supongamos que se tira un par de dados. Dejar\(X\) ser el número total de manchas que aparecen. Enrolle el par una\(X\) vez más. \(Y\)Sea el número de sietes que se lanzan en los\(X\) rollos. Determinar la distribución conjunta para el par\(\{X, Y\}\) y a partir de esto determinar la distribución marginal para\(Y\). ¿Cuál es la probabilidad de tres o más sietes?

    Contestar
    % file npr08_05.m
    % Data and basic calculations for Exercise 8.3.5.
    PX = (1/36)*[1 2 3 4 5 6 5 4 3 2 1];
    X = 2:12;
    Y = 0:12;
    P0 = zeros(11,13);
    for i = 1:11
      P0(i,1:i+2) = PX(i)*ibinom(i+1,1/6,0:i+1);
    end
    P = rot90(P0);
    PY = fliplr(sum(P'));
    disp('Answers are in X, Y, P, PY')
    npr08_05
    Answers are in X, Y, P, PY
    disp(PY)
      Columns 1 through 7
        0.3072    0.3660    0.2152    0.0828    0.0230    0.0048    0.0008
      Columns 8 through 13
        0.0001    0.0000    0.0000    0.0000    0.0000    0.0000

    Ejercicio\(\PageIndex{6}\)

    El par\(\{X, Y\}\) tiene la distribución conjunta (en 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]

    2020-04-11 11.56.15.png

    Determinar la distribución marginal y los valores de esquina para\(F_{XY}\). Determinar\(P(X + Y > 2)\) y\(P(X \ge Y)\).

    Contestar
    npr08_06
    Data are in X, Y, P
    jcalc
    Enter JOINT PROBABILITIES (as on the plane)  P
    Enter row matrix of VALUES of X  X
    Enter row matrix of VALUES of Y  Y
     Use array operations on matrices X, Y, PX, PY, t, u, and P
    disp([X;PX]')
       -2.3000    0.2300
       -0.7000    0.1700
        1.1000    0.2000
        3.9000    0.2020
        5.1000    0.1980
     
    disp([Y;PY]')
        1.3000    0.2980
        2.5000    0.3020
        4.1000    0.1900
        5.3000    0.2100
    jddbn
    Enter joint probability matrix (as on the plane)  P
    To view joint distribution function, call for FXY
    disp(FXY)
        0.2300    0.4000    0.6000    0.8020    1.0000
        0.1817    0.3160    0.4740    0.6361    0.7900
        0.1380    0.2400    0.3600    0.4860    0.6000
        0.0667    0.1160    0.1740    0.2391    0.2980
    P1 = total((t+u>2).*P)
    P1 =  0.7163
    P2 = total((t>=u).*P)
    P2 =  0.2799

    Ejercicio\(\PageIndex{7}\)

    El par\(\{X, Y\}\) tiene la distribución conjunta (en m-file npr08_07.m):

    \(P(X = i, Y = u)\)

    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

    Determinar las distribuciones marginales y los valores de esquina para\(F_{XY}\). Determinar\(P(1 \le X \le 4, Y > 4)\) y\(P(|X - Y| \le 2)\).

    Contestar
    npr08_07
    Data are in X, Y, P
    jcalc
    Enter JOINT PROBABILITIES (as on the plane)  P
    Enter row matrix of VALUES of X  X
    Enter row matrix of VALUES of Y  Y
     Use array operations on matrices X, Y, PX, PY, t, u, and P
    disp([X;PX]')
       -3.1000    0.1500
       -0.5000    0.2200
        1.2000    0.3300
        2.4000    0.1200
        3.7000    0.1100
        4.9000    0.0700
    disp([Y;PY]')
       -3.8000    0.1929
       -2.0000    0.3426
        4.1000    0.2706
        7.5000    0.1939
    jddbn
    Enter joint probability matrix (as on the plane)  P
    To view joint distribution function, call for FXY
    disp(FXY)
        0.1500    0.3700    0.7000    0.8200    0.9300    1.0000
        0.1410    0.3214    0.5920    0.6904    0.7564    0.8061
        0.0915    0.2719    0.4336    0.4792    0.5089    0.5355
        0.0510    0.0994    0.1720    0.1852    0.1852    0.1929
    M = (1<=t)&(t<=4)&(u>4);
    P1 = total(M.*P)
    P1 =  0.3230
    P2 = total((abs(t-u)<=2).*P)
    P2 =  0.3357

    Ejercicio\(\PageIndex{8}\)

    El par\(\{X, Y\}\) tiene la distribución conjunta (en m-file npr08_08.m):

    \(P(X = t, Y = u)\)

    t = 1 3 5 7 9 11 13 15 17 19
    u = 12 0.0156 0.0191 0.0081 0.0035 0.0091 0.0070 0.0098 0.0056 0.0091 0.0049
    10 0.0064 0.0204 0.0108 0.0040 0.0054 0.0080 0.0112 0.0064 0.0104 0.0056
    9 0.0196 0.0256 0.0126 0.0060 0.0156 0.0120 0.0168 0.0096 0.0056 0.0084
    5 0.0112 0.0182 0.0108 0.0070 0.0182 0.0140 0.0196 0.0012 0.0182 0.0038
    3 0.0060 0.0260 0.0162 0.0050 0.0160 0.0200 0.0280 0.0060 0.0160 0.0040
    -1 0.0096 0.0056 0.0072 0.0060 0.0256 0.0120 0.0268 0.0096 0.0256 0.0084
    -3 0.0044 0.0134 0.0180 0.0140 0.0234 0.0180 0.0252 0.0244 0.0234 0.0126
    -5 0.0072 0.0017 0.0063 0.0045 0.0167 0.0090 0.0026 0.0172 0.0217 0.0223

    Determinar las distribuciones marginales. Determinar\(F_{XY} (10, 6)\) y\(P(X > Y)\).

    Contestar
    npr08_08
    Data are in X, Y, P
    jcalc
    - - - - - - - - -
     Use array operations on matrices X, Y, PX, PY, t, u, and P
    disp([X;PX]')
        1.0000    0.0800
        3.0000    0.1300
        5.0000    0.0900
        7.0000    0.0500
        9.0000    0.1300
       11.0000    0.1000
       13.0000    0.1400
       15.0000    0.0800
       17.0000    0.1300
       19.0000    0.0700
    disp([Y;PY]')
       -5.0000    0.1092
       -3.0000    0.1768
       -1.0000    0.1364
        3.0000    0.1432
        5.0000    0.1222
        9.0000    0.1318
       10.0000    0.0886
       12.0000    0.0918
    F = total(((t<=10)&(u<=6)).*P)
    F =   0.2982
    P = total((t>u).*P)
    P =   0.7390

    Ejercicio\(\PageIndex{9}\)

    Se conservaron datos sobre el efecto del tiempo de capacitación en el tiempo para realizar un trabajo en una línea de producción. \(X\)es la cantidad de entrenamiento, en horas, y\(Y\) es el tiempo para realizar la tarea, en minutos. Los datos son los siguientes (en m-file npr08_09.m):

    \(P(X = t, Y = u)\)

    t = 1 1.5 2 2.5 3
    u = 5 0.039 0.011 0.005 0.001 0.001
    4 0.065 0.070 0.050 0.015 0.010
    3 0.031 0.061 0.137 0.051 0.033
    2 0.012 0.049 0.163 0.058 0.039
    1 0.003 0.009 0.045 0.025 0.017

    Determinar las distribuciones marginales. Determinar\(F_{XY}(2, 3)\) y\(P(Y/X \ge 1.25)\).

    Contestar
    npr08_09
    Data are in X, Y, P
    jcalc
    - - - - - - - - - - - -
     Use array operations on matrices X, Y, PX, PY, t, u, and P
    disp([X;PX]')
        1.0000    0.1500
        1.5000    0.2000
        2.0000    0.4000
        2.5000    0.1500
        3.0000    0.1000
    disp([Y;PY]')
        1.0000    0.0990
        2.0000    0.3210
        3.0000    0.3130
        4.0000    0.2100
        5.0000    0.0570
    F = total(((t<=2)&(u<=3)).*P)
    F =   0.5100
    P = total((u./t>=1.25).*P)
    P =   0.5570
    

    Para las densidades articulares en los Ejercicios 10-22 a continuación

    1. Esbozar la región de definición y determinar analíticamente las funciones de densidad marginal\(f_X\) y\(f_Y\).
    2. Utilice una aproximación discreta para trazar la densidad marginal\(f_X\) y la función de distribución marginal\(F_X\).
    3. Calcular analíticamente las probabilidades indicadas.
    4. Determinar por aproximación discreta las probabilidades indicadas.

    Ejercicio\(\PageIndex{10}\)

    \(f_{XY}(t, u) = 1\)para\(0 \le t \le 1\),\(0 \le u \le 2(1 - t)\).

    \(P(X > 1/2, Y > 1), P(0 \le X \le 1/2, Y > 1/2), P(Y \le X)\)

    Contestar

    Región es triángulo con vértices (0, 0), (1, 0), (0, 2).

    \(f_{X} (t) = \int_{0}^{2(1-t)} du = 2(1 - t)\),\(0 \le t \le 1\)

    \(f_{Y} (u) = \int_{0}^{1 - u/2} dt = 1 - u/2\),\(0 \le u \le 2\)

    \(M1 = \{(t, u):t > 1/2, u> 1\}\)yace fuera del trianlge\(P((X, Y) \in M1) = 0\)

    \(M2 = \{(t, u): 0 \le t \le 1/2, u > 1/2\}\)tiene área en el triángulo = 1/2

    \(M3\)= la región en el triángulo bajo\(u = t\), que tiene área 1/3

    tuappr
    Enter matrix [a b] of X-range endpoints  [0 1]
    Enter matrix [c d] of Y-range endpoints  [0 2]
    Enter number of X approximation points  200
    Enter number of Y approximation points  400
    Enter expression for joint density  (t<=1)&(u<=2*(1-t))
    Use array operations on X, Y, PX, PY, t, u, and P
    fx = PX/dx;
    FX = cumsum(PX);
    plot(X,fx,X,FX)          % Figure not reproduced
    M1 = (t>0.5)&(u>1);
    P1 = total(M1.*P)
    P1 =  0                  % Theoretical = 0
    M2 = (t<=0.5)&(u>0.5);
    P2 = total(M2.*P)
    P2 =  0.5000             % Theoretical = 1/2
    P3 = total((u<=t).*P)
    P3 =  0.3350             % Theoretical = 1/3

    Ejercicio\(\PageIndex{11}\)

    \(f_{XY} (t, u) = 1/2\)en el cuadrado con vértices en (1, 0), (2, 1), (1, 2), (0, 1).

    \(P(X > 1, Y > 1), P(X \le 1/2, 1 < Y), P(Y \le X)\)

    Contestar

    La región está delimitada por líneas\(u = 1 + t\),\( u = 1 - t\),\(u = 3 - t\), y\(u = t - 1\)

    \(f_X (t) = I_{[0,1]} (t) 0.5 \int_{1 - t}^{1 + t} du + I_{(1, 2]} (t) 0.5 \int_{t - 1}^{3 - t} du = I_{(1, 2]} (t) (2 - t) = f_Y(t)\)por simetría

    \(M1 = \{(t, u): t > 1, u > 1\}\)tiene área en el trangle = 1/2, entonces\(PM1 = 1/4\)

    \(M2 = \{(t, u): t \le 1/2, u > 1\}\)tiene área en el trangle = 1/8\), entonces\(PM2 = 1/16\)

    \(M3 = \{(t, u): u \le t\}\)tiene área en el trangle = 1, entonces\(PM3 = 1/2\)

    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  200
    Enter number of Y approximation points  200
    Enter expression for joint density  0.5*(u<=min(1+t,3-t))& ...
      (u>=max(1-t,t-1))
    Use array operations on X, Y, PX, PY, t, u, and P
    fx = PX/dx;
    FX = cumsum(PX);
    plot(X,fx,X,FX)          % Plot not shown
    M1 = (t>1)&(u>1);
    PM1 = total(M1.*P)
    PM1 =  0.2501            % Theoretical = 1/4
    M2 = (t<=1/2)&(u>1);
    PM2 = total(M2.*P)
    PM2 =  0.0631            % Theoretical = 1/16 = 0.0625
    M3 = u<=t;
    PM3 = total(M3.*P)
    PM3 =  0.5023            % Theoretical = 1/2

    Ejercicio\(\PageIndex{12}\)

    \(f_{XY} (t, u) = 4t(1 - u)\)para\(0 \le t \le 1\),\(0 \le u \le 1\).

    \(P(1/2 < X < 3/4, Y > 1/2)\),\(P(X \le 1/2, Y > 1/2)\),\(P(Y \le X)\)

    Contestar

    Región es la unidad cuadrada,

    \(f_X (t) = \int_{0}^{1} 4t(1 - u) du = 2t\),\(0 \le t \le 1\)

    \(f_Y(u) = \int_{0}^{1} 4t(1 - u) dt = 2(1 - u)\),\(0 \le u \le 1\)

    \(P1 = \int_{1/2}^{3/4} \int_{1/2}^{1} 4t (1 - u) du dt = 5/64\)\(P2 = \int_{0}^{1/2} \int_{1/2}^{1} 4t(1 - u) dudt = 1/16\)

    \(P3 = \int_{0}^{1} \int_{0}^{t} 4t(1 - u) du dt = 5/6\)

    tuappr
    Enter matrix [a b] of X-range endpoints  [0 1]
    Enter matrix [c d] of Y-range endpoints  [0 1]
    Enter number of X approximation points  200
    Enter number of Y approximation points  200
    Enter expression for joint density  4*t.*(1 - u)
    Use array operations on X, Y, PX, PY, t, u, and P
    fx = PX/dx;
    FX = cumsum(PX);
    plot(X,fx,X,FX)           % Plot not shown
    M1 = (1/2<t)&(t<3/4)&(u>1/2);
    P1 = total(M1.*P)
    P1 =  0.0781              % Theoretical = 5/64 = 0.0781
    M2 = (t<=1/2)&(u>1/2);
    P2 = total(M2.*P)
    P2 =  0.0625              % Theoretical = 1/16 = 0.0625
    M3 = (u<=t);
    P3 = total(M3.*P)
    P3 =  0.8350              % Theoretical = 5/6 = 0.8333

    Ejercicio\(\PageIndex{13}\)

    \(f_{XY} (t, u) = \dfrac{1}{8} (t + u)\)para\(0 \le t \le 2\),\(0 \le u \le 2\).

    \(P(X > 1/2, Y > 1/2), P(0 \le X \le 1, Y > 1), P(Y \le X)\)

    Contestar

    Región es la plaza\(0 \le t \le 2\),\(0 \le u \le 2\)

    \(f_X (t) = \dfrac{1}{8} \int_{0}^{2} (t + u) = \dfrac{1}{4} ( t + 1) = f_Y(t)\),\(0 \le t \le 2\)

    \(P1 = \int_{1/2}^{2} \int_{1/2}^{2} (t + u) dudt = 45/64\)\(P2 = \int_{0}^{1} \int_{1}^{2} (t + u) du dt = 1/4\)

    \(P3 = \int_{0}^{2} \int_{0}^{1} (t + u) dudt = 1/2\)

    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  200
    Enter number of Y approximation points  200
    Enter expression for joint density  (1/8)*(t+u)
    Use array operations on X, Y, PX, PY, t, u, and P
    fx = PX/dx;
    FX = cumsum(PX);
    plot(X,fx,X,FX)
    M1 = (t>1/2)&(u>1/2);
    P1 = total(M1.*P)
    P1 =  0.7031              % Theoretical = 45/64 = 0.7031
    M2 = (t<=1)&(u>1);
    P2 = total(M2.*P)
    P2 =  0.2500              % Theoretical = 1/4
    M3 = u<=t;
    P3 = total(M3.*P)
    P3 =  0.5025              % Theoretical = 1/2

    Ejercicio\(\PageIndex{14}\)

    \(f_{XY}(t, u) = 4ue^{-2t}\)para\(0 \le t, 0 \le u \le 1\)

    \(P(X \le 1, Y > 1), P(X > 0, 1/2 < Y < 3/4), P(X < Y)\)

    Contestar

    La región es despojada\(t = 0, u = 0, u = 1\)

    \(f_X(t) = 2e^{-2t}\),\(0 \le t\),\(f_Y(u) = 2u\),\(0 \le u \le 1\),\(f_{XY} = f_X f_Y\)

    \(P1 = 0\),\(P2 = \int_{0.5}^{\infty} 2e^{-2t} dt \int_{1/2}^{3/4} 2udu = e^{-1} 5/16\)

    \(P3 = 4 \int_{0}^{1} \int_{t}^{1} ue^{-2t} dudt = \dfrac{3}{2} e^{-2} + \dfrac{1}{2} = 0.7030\)

    tuappr
    Enter matrix [a b] of X-range endpoints  [0 3]
    Enter matrix [c d] of Y-range endpoints  [0 1]
    Enter number of X approximation points  400
    Enter number of Y approximation points  200
    Enter expression for joint density  4*u.*exp(-2*t)
    Use array operations on X, Y, PX, PY, t, u, and P
    M2 = (t > 0.5)&(u > 0.5)&(u<3/4);
    p2 = total(M2.*P)
    p2 =  0.1139            % Theoretical = (5/16)exp(-1) = 0.1150
    p3 = total((t<u).*P)
    p3 =  0.7047            % Theoretical = 0.7030

    Ejercicio\(\PageIndex{15}\)

    \(f_{XY} (t, u) = \dfrac{3}{88} (2t + 3u^2)\)para\(0 \le t \le 2\),\(0 \le u \le 1 + t\).

    \(F_{XY} (1, 1)\),\(P(X \le 1, Y > 1)\),\(P(|X - Y| < 1)\)

    Contestar

    Región delimitada por\(t = 0\)\(t = 2\),\(u = 0\),\(u = 1 + t\)

    \(f_X (t) = \dfrac{3}{88} \int_{0}^{1 + t} (2t + 3u^2) du = \dfrac{3}{88}(1 + t)(1 + 4t + t^2) = \dfrac{3}{88} ( 1 + 5t + 5t^2 + t^3)\),\(0 \le t \le 2\)

    \(f_Y(u) = I_{[0,1]} (u) \dfrac{3}{88} \int_{0}^{2} (2t + 3u^2) dt + I_{(1, 3]} (u) \dfrac{3}{88} \int_{u - 1}^{2} (2t + 3u^2) dt = \)

    \(I_{[0,1]} (u) \dfrac{3}{88} (6u^2 + 4) + I_{(1,3]} (t) \dfrac{3}{88} (3 + 2u + 8u^2 - 3u^3)\)

    \(F_{XY}(1, 1) = \int_{0}^{1} \int_{0}^{1} f_{XY} (t, u) dudt = 3/44\)

    \(P1 = \int_{0}^{1} \int_{1}^{1 + t} f_{XY} (t, u)dudt = 41/352\)\(P2 = \int_{0}^{1} \int_{1}^{1 + t} f_{XY} (t, u) dudt = 329/352\)

    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  200
    Enter 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 P
    fx = PX/dx;
    FX = cumsum(PX);
    plot(X,fx,X,FX)
    MF = (t<=1)&(u<=1);
    F = total(MF.*P)
    F =   0.0681            % Theoretical = 3/44 = 0.0682
    M1 = (t<=1)&(u>1);
    P1 = total(M1.*P)
    P1 =  0.1172            % Theoretical = 41/352 = 0.1165
    M2 = abs(t-u)<1;
    P2 = total(M2.*P)
    P2 =  0.9297           % Theoretical = 329/352 = 0.9347

    Ejercicio\(\PageIndex{16}\)

    \(f_{XY} (t, u) = 12t^2u\)en el paralelogramo con vértices (-1, 0), (0, 0), (1, 1), (0, 1).

    \(P(X \le 1/2, Y > 0), P(X < 1/2, Y \le 1/2), P(Y \ge 1/2)\)

    Contestar

    Región delimitada por\(u = 0\)\(u = t\),\(u = 1\),\(u = t + 1\)

    \(f_X (t) = I_{[-1, 0]} (t) 12 \int_{0}^{t + 1} t^2 u du + I_{(0, 1]} (t) 12 \int_{t}^{1} t^2 u du = I_{[-1, 0]} (t) 6t^2 (t + 1)^2 + I_{(0, 1]}(t) 6t^2(1 - t^2)\)

    \(f_Y(u) = 12\int_{u - 1}^{t} t^2 udu + 12u^3 - 12u^2 + 4u\),\(0 \le u \le 1\)

    \(P1 = 1 - 12 \int_{1/2}^{1} \int_{t}^{1} t^2 ududt = 33/80\),\(P2 = 12 \int_{0}^{1/2} \int_{u - 1}^{u} t^2 udtdu = 3/16\)

    \(P3 = 1 - P2 = 13/16\)

    tuappr
    Enter matrix [a b] of X-range endpoints  [-1 1]
    Enter matrix [c d] of Y-range endpoints  [0 1]
    Enter number of X approximation points  400
    Enter number of Y approximation points  200
    Enter expression for joint density  12*u.*t.^2.*((u<=t+1)&(u>=t))
    Use array operations on X, Y, PX, PY, t, u, and P
    p1 = total((t<=1/2).*P)
    p1 =  0.4098                % Theoretical = 33/80 = 0.4125
    M2 = (t<1/2)&(u<=1/2);
    p2 = total(M2.*P)
    p2 =  0.1856                % Theoretical = 3/16  = 0.1875
    P3 = total((u>=1/2).*P)
    P3 =  0.8144                % Theoretical = 13/16 = 0.8125

    Ejercicio\(\PageIndex{17}\)

    \(f_{XY} (t, u) = \dfrac{24}{11} tu\)para\(0 \le t \le 2\),\(0 \le u \le \text{min}\ \{1, 2 - t\}\)

    \(P(X \le 1, Y \le 1), P(X > 1), P(X < Y)\)

    Contestar

    La región está delimitada por\(t = 0, u = 0, u = 2, u = 2 - t\)

    \(f_X (t) = I_{[0, 1]} (t) \dfrac{24}{11} \int_{0}^{1} tudu + I_{(1, 2]} (t) \dfrac{24}{11} \int_{0}^{2 - t} tudu =\)

    \(I_{[0, 1]} (t) \dfrac{12}{11} t + I_{(1, 2]} (t) \dfrac{12}{11} t(2 - t)^2\)

    \(f_Y (u) = \dfrac{24}{11} \int_{0}^{2 - u} tudt = \dfrac{12}{11} u(u - 2)^2\),\(0 \le u \le 1\)

    \(P1 = \dfrac{24}{11} \int_{0}^{1} \int_{0}^{1} tududt = 6/11\)\(P2 = \dfrac{24}{11} \int_{1}^{2} \int_{0}^{2 - t} tududt = 5/11\)

    \(P3 = \dfrac{24}{11} \int_{0}^{1} \int_{t}^{1} tududt = 3/11\)

    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  400
    Enter number of Y approximation points  200
    Enter expression for joint density  (24/11)*t.*u.*(u<=2-t)
    Use array operations on X, Y, PX, PY, t, u, and P
    M1 = (t<=1)&(u<=1);
    P1 = total(M1.*P)
    P1 = 0.5447             % Theoretical = 6/11 = 0.5455
    P2 = total((t>1).*P)
    P2 =  0.4553            % Theoretical = 5/11 = 0.4545
    P3 = total((t<u).*P)
    P3 =  0.2705            % Theoretical = 3/11 = 0.2727

    Ejercicio\(\PageIndex{18}\)

    \(f_{XY} (t, u) = \dfrac{3}{23} (t + 2u)\)para\(0 \le t \le 2\),\(0 \le u \le \text{max}\ \{2 - t, t\}\)

    \(P(X \ge 1, Y \ge 1), P(Y \le 1), P(Y \le X)\)

    Contestar

    La región está delimitada por\(t = 0, t = 2, u = 0, u = 2 - t\)\((0 \le t \le 1)\),\(u = t (1 < t \le 2)\)

    \(f_X(t) = I_{[0,1]} (t) \dfrac{3}{23} \int_{0}^{2 - t} (t + 2u) du + I_{(1, 2]} (t) \dfrac{3}{23} \int_{0}^{t} (t + 2u) du = I_{[0, 1]} (t) \dfrac{6}{23} (2 - t) + I_{(1, 2]} (t) \dfrac{6}{23}t^2\)

    \(f_Y(u) = I_{[0, 1]} (u) \dfrac{3}{23} \int_{0}^{2} (t + 2u) du + I_{(1, 2]} (u) [\dfrac{3}{23} \int_{0}^{2 - u} (t + 2u) dt + \dfrac{3}{23} \int_{u}^{2} (t + 2u) dt]=\)

    \(I_{[0,1]} (u) \dfrac{6}{23} (2u + 1) + I_{(1, 2]} (u) \dfrac{3}{23} (4 + 6u - 4u^2)\)

    \(P1 = \dfrac{3}{23} \int_{1}^{2} \int_{1}^{t} (t + 2u) du dt = 13/46\),\(P2 = \dfrac{3}{23} \int_{0}^{2} \int_{0}^{1} (t + 2u) du dt = 12/23\)

    \(P3 = \dfrac{3}{23} \int_{0}^{2} \int_{0}^{t} (t + 2u) dudt = 16/23\)

    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  200
    Enter number of Y approximation points  200
    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 P
    M1 = (t>=1)&(u>=1);
    P1 = total(M1.*P)
    P1 =  0.2841
    13/46                 % Theoretical = 13/46 = 0.2826
    P2 = total((u<=1).*P)
    P2 =  0.5190             % Theoretical = 12/23 = 0.5217
    P3 = total((u<=t).*P)
    P3 =  0.6959             % Theoretical = 16/23 = 0.6957

    Ejercicio\(\PageIndex{19}\)

    \(f_{XY} (t, u) = \dfrac{12}{179} (3t^2 + u)\), para\(0 \le t \le 2\),\(0 \le u \le \text{min } \{1 + t, 2\}\)

    \(P(X \ge 1, Y \ge 1), P(X \le 1, Y \le 1), P(Y < X)\)

    Contestar

    La región tiene dos partes: (1)\(0 \le t \le 1, 0 \le u \le 2\) (2)\(1 < t \le 2, 0 \le u \le 3 - t\)

    \(f_X (t) = I_{[0, 1]} (t) \dfrac{12}{179} \int_{0}^{2} (3t^2 + u) du + I_{(1, 2]} (t) \dfrac{12}{179} \int_{0}^{3 - t} (3t^2 + u) du =\)

    \(I_{[0, 1]} (t) \dfrac{24}{179} (3t^2 + 1) + I_{(1, 2]} (t) \dfrac{6}{179} (9 - 6t + 19t^2 - 6t^3)\)

    \(f_Y(u) = I_{[0, 1]} (u) \dfrac{12}{179} \int_{0}^{2}(3t^2 + u) dt + I_{(1, 2]} (u) \dfrac{12}{179} \int_{0}^{3 - u} (3t^2 + u) dt =\)

    \(I_{[0, 1]} (u) \dfrac{24}{179} (4 + u) + I_{(1, 2]} (u) \dfrac{12}{179} (27 - 24u + 8u^2 - u^3)\)

    \(P1 = \dfrac{12}{179} \int{1}^{2} \int_{1}^{3 - t} (3t^2 + u) du dt = 41/179\)\(P2 = \dfrac{12}{179} \int_{0}^{1} \int_{0}^{1} (3t^2 + u) dudt = 18/179\)

    \(P3 = \dfrac{12}{179} \int_{0}^{3/2} \int_{0}^{t} (3t^2 + u) dudt + \dfrac{12}{179} \int_{3/2}^{2} \int_{0}^{3 - t} (3t^2 + u) dudt = 1001/1432\)

    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  200
    Enter number of Y approximation points  200
    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 P
    fx = PX/dx;
    FX = cumsum(PX);
    plot(X,fx,X,FX)
    M1 = (t>=1)&(u>=1);
    P1 = total(M1.*P)
    P1 =  2312            % Theoretical = 41/179 = 0.2291
    M2 = (t<=1)&(u<=1);
    P2 = total(M2.*P)
    P2 =  0.1003           % Theoretical = 18/179 = 0.1006
    M3 = u<=min(t,3-t);
    P3 = total(M3.*P)
    P3 =  0.7003            % Theoretical = 1001/1432 = 0.6990

    Ejercicio\(\PageIndex{20}\)

    \(f_{XY} (t, u) = \dfrac{12}{227} (3t + 2tu)\)para\(0 \le t \le 2\),\(0 \le u \le \text{min} \{1 + t, 2\}\)

    \(P(X \le 1/2, Y \le 3/2), P(X \le 1.5, Y > 1), P(Y < X)\)

    Contestar

    La región se divide en dos partes:

    1. \(0 \le t \le 1\),\(0 \le u \le 1 + t\)
    2. \(1 < t \le 2\),\(0 \le u \le 2\)

    \(f_X(t) = I_{[0,1]} (t) \int_{0}^{1+t} f_{XY} (t, u) du + I_{(1, 2]} (t) \int_{0}^{2} f_{XY} (t, u) du =\)

    \(I_{[0, 1]} (t) \dfrac{12}{227} (t^3 + 5t^2 + 4t) + I_{(1, 2]} (t) \dfrac{120}{227} t\)

    \(f_Y(u) = I_{[0, 1]} (u) \int_{0}^{2} f_{XY} (t, u) dt + I_{(1, 2]} (u) \int_{u - 1}^{2} f_{XY} (t, u) dt = \)

    \(I_{[0, 1]} (u) \dfrac{24}{227} (2u + 3) + I_{(1, 2]} (u) \dfrac{6}{227} (2u + 3) (3 + 2u - u^2)\)

    \(= I_{[0, 1]} (u) \dfrac{24}{227} (2u + 3) + I_{(1, 2]} (u) \dfrac{6}{227} (9 + 12 u + u^2 - 2u^3)\)

    \(P1 = \dfrac{12}{227} \int_{0}^{1/2} \int_{0}^{1 + t} (3t + 2tu) du dt = 139/3632\)

    \(P2 = \dfrac{12}{227} \int_{0}^{1} \int_{1}^{1 + t} (3t + 2tu) dudt + \dfrac{12}{227} \int_{1}^{3/2} \int_{1}^{2} (3t + 2tu) du dt = 68/227\)

    \(P3 = \dfrac{12}{227} \int_{0}^{2} \int_{1}^{t} (3t + 2tu) dudt = 144/227\)

    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  200
    Enter number of Y approximation points  200
    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 P
    M1 = (t<=1/2)&(u<=3/2);
    P1 = total(M1.*P)
    P1 =  0.0384             % Theoretical = 139/3632 = 0.0383
    M2 = (t<=3/2)&(u>1);
    P2 = total(M2.*P)
    P2 =  0.3001             % Theoretical = 68/227 = 0.2996
    M3 = u<t;
    P3 = total(M3.*P)
    P3 =  0.6308             % Theoretical = 144/227 = 0.6344

    Ejercicio\(\PageIndex{21}\)

    \(f_{XY} (t, u) = \dfrac{2}{13} (t + 2u)\)para\(0 \le t \le 2\),\(0 \le u \le \text{min}\ \{2t, 3 - t\}\)

    \(P(X < 1), P(X \ge 1, Y \le 1), P(Y \le X/2)\)

    Contestar

    Región delimitada por\(t = 2, u = 2t\)\((0 \le t \le 1)\),\(3 - t\)\((1 \le t \le 2)\)

    \(f_X(t) = I_{[0, 1]} (t) \dfrac{2}{13} \int_{0}^{2t} (t + 2u) du + I_{(1, 2]} (t) \dfrac{2}{13} \int_{0}^{3 - t} (t + 2u) du = I_{[0, 1]} (t) \dfrac{12}{13} t^2 + I_{(1, 2]} (t) \dfrac{6}{13} (3 - t)\)

    \(f_Y (u) = I_{[0, 1]} (u) \dfrac{2}{13} \int_{u/2}^{2} (t + 2u) dt + I_{(1, 2]} (u) \dfrac{2}{13} \int_{u/2}^{3 - u} (t + 2u) dt =\)

    \(I_{[0, 1]} (u) (\dfrac{4}{13} + \dfrac{8}{13}u - \dfrac{9}{52} u^2) + I_{(1, 2]} (u) (\dfrac{9}{13} + \dfrac{6}{13} u - \dfrac{21}{52} u^2)\)

    \(P1 = \int_{0}^{1} \int_{0}^{2t} (t + 2u) dudt = 4/13\)\(P2 = \int_{1}^{2} \int_{0}^{1} (t + 2u)dudt = 5/13\)

    \(P3 = \int_{0}^{2} \int_{0}^{u/2} (t + 2u) dudt = 4/13\)

    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  400
    Enter number of Y approximation points  400
    Enter expression for joint density  (2/13)*(t+2*u).*(u<=min(2*t,3-t))
    Use array operations on X, Y, PX, PY, t, u, and P
    P1 = total((t<1).*P)
    P1 = 0.3076             % Theoretical = 4/13 = 0.3077
    M2 = (t>=1)&(u<=1);
    P2 = total(M2.*P)
    P2 =  0.3844            % Theoretical = 5/13 = 0.3846
    P3 = total((u<=t/2).*P)
    P3 =  0.3076             % Theoretical = 4/13 = 0.3077

    Ejercicio\(\PageIndex{22}\)

    \(f_{XY} (t, u) = I_{[0, 1]} (t) \dfrac{3}{8} (t^2 + 2u) + I_{(1, 2]} (t) \dfrac{9}{14} t^2u^2\)para\(0 \le u \le 1\).

    \(P(1/2 \le X \le 3/2, Y \le 1/2)\)

    Contestar

    Región es rectángulo delimitado por\(t = 0\),\(t = 2\),\(u = 0\),\(u = 1\)

    \(f_{XY} (t, u) = I_{[0, 1]} (t) \dfrac{3}{8} (t^2 + 2u) + I_{(1, 2]} (t) \dfrac{9}{14} t^2 u^2\),\(0 \le u \le 1\)

    \(f_X (t) = I_{[0, 1]} (t) \dfrac{3}{8} \int_{0}^{1} (t^2 + 2u) du + I_{(1, 2]} (t) \dfrac{9}{14} \int_{0}^{1} t^2 u^2 du = I_{[0,1]} (t) \dfrac{3}{8} (t^2 + 1) + I_{(1, 2]} (t) \dfrac{3}{14} t^2\)

    \(f_Y(u) = \dfrac{3}{8} \int_{0}^{1} (t^2 + 2u0 dt + \dfrac{9}{14} \int_{1}^{2} t^2 u^2 dt = \dfrac{1}{8} + \dfrac{3}{4} u + \dfrac{3}{2} u^2\)\(0 \le u \le 1\)

    \(P1 = \dfrac{3}{8} \int_{1/2}^{1} \int_{0}^{1/2} (t^2 + 2u) dudt + \dfrac{9}{14} \int_{1}^{3/2} \int_{0}^{1/2} t^2 u^2 dudt = 55/448\)

    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  400
    Enter number of Y approximation points  200
    Enter expression for joint density  (3/8)*(t.^2+2*u).*(t<=1) ...
           + (9/14)*(t.^2.*u.^2).*(t > 1)
    Use array operations on X, Y, PX, PY, t, u, and P
    M = (1/2<=t)&(t<=3/2)&(u<=1/2);
    P = total(M.*P)
    P =  0.1228          % Theoretical = 55/448 = 0.1228
    

    This page titled 8.3: Problemas en Vectores Aleatorios y Distribuciones Conjuntas is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Paul Pfeiffer via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.