8.2 Problems on random vectors and joint distributions (Page 5/6)

Applied probability Page 5 / 6

$f_{X Y} (t, u) = \frac{1}{8} (t + u)$ for $0 \leq t \leq 2$ , $0 \leq u \leq 2$ .

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

Region is the square $0 \leq t \leq 2, 0 \leq u \leq 2$ .

f_{X} (t) = \frac{1}{8} \int_{0}^{2} (t + u) = \frac{1}{4} (t + 1) = f_{Y} (t), 0 \leq t \leq 2

P 1 = \int_{1 / 2}^{2} \int_{1 / 2}^{2} (t + u) d u d t = 45 / 64 P 2 = \int_{0}^{1} \int_{1}^{2} (t + u) d u d t = 1 / 4

P 3 = \int_{0}^{2} \int_{0}^{t} (t + u) d u d t = 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  200Enter 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

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$f_{X Y} (t, u) = 4 u e^{- 2 t}$ for $0 \leq t, 0 \leq u \leq 1$

P (X \leq 1, Y > 1), P (X > 0.5, 1 / 2 < Y < 3 / 4), P (X < Y)

Region is strip bounded by $t = 0, u = 0, u = 1$

f_{X} (t) = 2 e^{- 2 t}, 0 \leq t, f_{Y} (u) = 2 u, 0 \leq u \leq 1, f_{X Y} = f_{X} f_{Y}

P 1 = 0, P 2 = \int_{0.5}^{\infty} 2 e^{- 2 t} d t \int_{1 / 2}^{3 / 4} 2 u d u = e^{- 1} 5 / 16

P 3 = 4 \int_{0}^{1} \int_{t}^{1} u e^{- 2 t} d u d t = \frac{3}{2} e^{- 2} + \frac{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  400Enter 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

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$f_{X Y} (t, u) = \frac{3}{88} (2 t + 3 u^{2})$ for $0 \leq t \leq 2$ , $0 \leq u \leq 1 + t$ .

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

Region bounded by $t = 0, t = 2, u = 0, u = 1 + t$

f_{X} (t) = \frac{3}{88} \int_{0}^{1 + t} (2 t + 3 u^{2}) d u = \frac{3}{88} (1 + t) (1 + 4 t + t^{2}) = \frac{3}{88} (1 + 5 t + 5 t^{2} + t^{3}), 0 \leq t \leq 2

f_{Y} (u) = I_{[0, 1]} (u) \frac{3}{88} \int_{0}^{2} (2 t + 3 u^{2}) d t + I_{(1, 3]} (u) \frac{3}{88} \int_{u - 1}^{2} (2 t + 3 u^{2}) d t =

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

F_{X Y} (1, 1) = \int_{0}^{1} \int_{0}^{1} f_{X Y} (t, u) d u d t = 3 / 44

P 1 = \int_{0}^{1} \int_{1}^{1 + t} f_{X Y} (t, u) d u d t = 41 / 352 P 2 = \int_{0}^{1} \int_{1}^{1 + t} f_{X Y} (t, u) d u d t = 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  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 Pfx = 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

Got questions? Get instant answers now!

$f_{X Y} (t, u) = 12 t^{2} u$ on the parallelogram with vertices $(- 1, 0), (0, 0), (1, 1), (0, 1)$

P (X \leq 1 / 2, Y > 0), P (X < 1 / 2, Y \leq 1 / 2), P (Y \geq 1 / 2)

Region bounded by $u = 0, u = t, u = 1, u = t + 1$

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

f_{Y} (u) = 12 \int_{u - 1}^{t} t^{2} u d t + 12 u^{3} - 12 u^{2} + 4 u, 0 \leq u \leq 1

P 1 = 1 - 12 \int_{1 / 2}^{1} \int_{t}^{1} t^{2} u d u d t = 33 / 80, P 2 = 12 \int_{0}^{1 / 2} \int_{u - 1}^{u} t^{2} u d t d u = 3 / 16

P 3 = 1 - P 2 = 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  400Enter 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 Pp1 = total((t<=1/2).*P)
p1 =  0.4098                % Theoretical = 33/80 = 0.4125M2 = (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

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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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