8.2 Problems on random vectors and joint distributions (Page 4/6)

Applied probability Page 4 / 6

For the joint densities in Exercises 10-22 below

Sketch the region of definition and determine analytically the marginal density functions f _X and f _Y .
Use a discrete approximation to plot the marginal density f _X and the marginal distribution function F _X .
Calculate analytically the indicated probabilities.
Determine by discrete approximation the indicated probabilities.

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

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

Region is triangle with vertices (0,0), (1,0), (0,2).

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

f_{Y} (u) = \int_{0}^{1 - u / 2} d t = 1 - u / 2, 0 \leq u \leq 2

M 1 = {(t, u) : t > 1 / 2, u > 1} lies outside the triangle P ((X, Y) \in M 1) = 0

M 2 = {(t, u) : 0 \leq t \leq 1 / 2, u > 1 / 2} has area in the trangle = 1 / 2

M 3 = the region in the triangle under u = t, which has area 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  200Enter 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 Pfx = 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

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$f_{X Y} (t, u) = 1 / 2$ on the square with vertices at $(1, 0), (2, 1), (1, 2), (0, 1)$ .

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

The region is bounded by the lines $u = 1 + t$ , $u = 1 - t$ , $u = 3 - t$ , and $u = t - 1$

f_{X} (t) = I_{[0, 1]} (t) 0.5 \int_{1 - t}^{1 + t} d u + I_{(1, 2]} (t) 0.5 \int_{t - 1}^{3 - t} d u = I_{[0, 1]} (t) t + I_{(1, 2]} (t) (2 - t) = f_{Y} (t) by symmetry

M 1 = {(t, u) : t > 1, u > 1} has area in the trangle = 1 / 2, so P M 1 = 1 / 4

M 2 = {(t, u) : t \leq 1 / 2, u > 1} has area in the trangle = 1 / 8, so P M 2 = 1 / 16

M 3 = {(t, u) : u \leq t} has area in the trangle = 1, so P M 3 = 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  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 Pfx = 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

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

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

Region is the unit square.

f_{X} (t) = \int_{0}^{1} 4 t (1 - u) d u = 2 t, 0 \leq t \leq 1

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

P 1 = \int_{1 / 2}^{3 / 4} \int_{1 / 2}^{1} 4 t (1 - u) d u d t = 5 / 64 P 2 = \int_{0}^{1 / 2} \int_{1 / 2}^{1} 4 t (1 - u) d u d t = 1 / 16

P 3 = \int_{0}^{1} \int_{0}^{t} 4 t (1 - u) d u d t = 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  200Enter 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 shownM1 = (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

Got questions? Get instant answers now!

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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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