14.2: Problems on Conditional Expectation, Regression

For the distributions in Exercises 1-3

1. Determine the regression curve of \(Y\) on \(X\) and compare with the regression line of \(Y\) on \(X\).
2. For the function \(Z = g(X, Y)\) indicated in each case, determine the regression curve of \(Z\) on \(X\).

Exercise \(\PageIndex{1}\)

(See Exercise 17 from "Problems on Mathematical Expectation"). The pair \(\{X, Y\}\) has the joint distribution (in file npr08_07.m):

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

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

The regression line of \(Y\) on \(X\) is \(u = 0.5275 t + 0.6924\).

\(Z = X^2Y + |X + Y|\)

Answer

The regression line of \(Y\) on \(X\) is \(u = 0.5275t + 0.6924\).

npr08_07
Data are in X, Y, P
jcalc
- - - - - - - - - - -
EYx = sum(u.*P)./sum(P);
disp([X;EYx]')
   -3.1000   -0.0290
   -0.5000   -0.6860
    1.2000    1.3270
    2.4000    2.1960
    3.7000    3.8130
    4.9000    2.5700
G = t.^2.*u + abs(t+u);
EZx = sum(G.*P)./sum(P);
disp([X;EZx]')
   -3.1000    4.0383
   -0.5000    3.5345
    1.2000    6.0139
    2.4000   17.5530
    3.7000   59.7130
    4.9000   69.1757

Exercise \(\PageIndex{2}\)

(See Exercise 18 from "Problems on Mathematical Expectation"). The pair \(\{X, Y\}\) has the joint distribution (in 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

The regression line of \(Y\) on \(X\) is \(u = -0.2584 t + 5.6110\).

\(Z = I_Q (X, Y) \sqrt{X} (Y - 4) + I_{Q^c} (X, Y) XY^2\) \(Q = \{(t, u) : u \le t \}\)

Answer

The regression line of \(Y\) on \(X\) is \(u = -0.2584 t + 5.6110\).

npr08_08
Data are in X, Y, P
jcalc
- - - - - - - - - - - -
EYx = sum(u.*P)./sum(P);
disp([X;EYx]')
    1.0000    5.5350
    3.0000    5.9869
    5.0000    3.6500
    7.0000    2.3100
    9.0000    2.0254
   11.0000    2.9100
   13.0000    3.1957
   15.0000    0.9100
   17.0000    1.5254
   19.0000    0.9100
M = u<=t;
G = (u-4).*sqrt(t).*M + t.*u.^2.*(1-M);
EZx = sum(G.*P)./sum(P);
disp([X;EZx]')
    1.0000   58.3050
    3.0000  166.7269
    5.0000  175.9322
    7.0000  185.7896
    9.0000  119.7531
   11.0000  105.4076
   13.0000   -2.8999
   15.0000  -11.9675
   17.0000  -10.2031
   19.0000  -13.4690

Exercise \(\PageIndex{3}\)

(See Exercise 19 from "Problems on Mathematical Expectation"). Data were kept on the effect of training time on the time to perform a job on a production line. \(X\) is the amount of training, in hours, and \(Y\) is the time to perform the task, in minutes. The data are as follows (in 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

The regression line of \(Y\) on \(X\) is \(u = -0.7793t + 4.3051\).

\(Z = (Y -2.8)/X\)

Answer

The regression line of \(Y\) on \(X\) is \(u = -0.7793t + 4.3051\).

npr08_09
Data are in X, Y, P
jcalc
- - - - - - - - - - - -
EYx = sum(u.*P)./sum(P);
disp([X;EYx]')
    1.0000    3.8333
    1.5000    3.1250
    2.0000    2.5175
    2.5000    2.3933
    3.0000    2.3900
G = (u - 2.8)./t;
EZx = sum(G.*P)./sum(P);
disp([X;EZx]')
    1.0000    1.0333
    1.5000    0.2167
    2.0000   -0.1412
    2.5000   -0.1627
    3.0000   -0.1367

For the joint densities in Exercises 4-11 below

1. Determine analytically the regression curve of \(Y\) on \(X\) and compare with the regression line of \(Y\) on \(X\).
2. Check these with a discrete approximation.

Exercise \(\PageIndex{4}\)

(See Exercise 10 from "Problems On Random Vectors and Joint Distributions", Exercise 20 from "Problems on Mathematical Expectation", and Exercise 23 from "Problems on Variance, Covariance, Linear Regression"). \(f_{XY} (t, u) = 1\) for \(0 \le t \le 1\). \(0 \le u \le 2(1 - t)\).

The regression line of \(Y\) on \(X\) is \(u = 1 - t\).

\(f_X (t) = 2(1 - t)\), \(0 \le t \le 1\)

Answer

The regression line of \(Y\) on \(X\) is \(u = 1 - t\).

\(f_{Y|X} (u|t) = \dfrac{1}{2(1 - t)}\). \(0 \le t \le 1\), \(0 \le u \le 2(1 - t)\)

\(E[Y|X = t] = \dfrac{1}{2(1 - t)} \int_{0}^{2(1-t)} udu = 1 - t\), \(0 \le t \le 1\)

tuappr: [0 1] [0 2] 200 400 u<=2*(1-t)
- - - - - - - - - - - - -
EYx = sum(u.*P)./sum(P);
plot(X,EYx)   % Straight line thru  (0,1), (1,0)

Exercise \(\PageIndex{5}\)

(See Exercise 13 from " Problems On Random Vectors and Joint Distributions", Exercise 23 from "Problems on Mathematical Expectation", and Exercise 24 from "Problems on Variance, Covariance, Linear Regression"). \(f_{XY} (t, u) = \dfrac{1}{8} (t+u)\) for \(0 \le t \le 2\), \(0 \le u \le 2\).

The regression line of \(Y\) on \(X\) is \(u = -t/11 + 35/33\).

\(f_{X} (t) = \dfrac{1}{4} (t + 1)\), \(0 \le t \le 2\)

Answer

The regression line of \(Y\) on \(X\) is \(u = -t/11 + 35/33\).

\(f_{Y|X} (u|t) = \dfrac{(t + u)}{2(t + 1)}\) \(0 \le t \le 2\), \(0 \le u \le 2\)

\(E[Y|X = t] = \dfrac{1}{2(t + 1)} \int_{0}^{2} (tu + u^2)\ du = 1 + \dfrac{1}{3t+3}\) \(0 \le t \le 2\)

tuappr: [0 2] [0 2] 200 200 (1/8)*(t+u)
EYx = sum(u.*P)./sum(P);
eyx = 1 + 1./(3*X+3);
plot(X,EYx,X,eyx)            % Plots nearly indistinguishable

Exercise \(\PageIndex{6}\)

(See Exercise 15 from " Problems On Random Vectors and Joint Distributions", Exercise 25 from "Problems on Mathematical Expectation", and Exercise 25 from "Problems on Variance, Covariance, Linear Regression"). \(f_{XY} (t, u) = \dfrac{3}{88} (2t + 3u^2)\) for \(0 \le t \le 2\), \(0 \le u \le 1 + t\).

The regression line of \(Y\) on \(X\) is \(u = 0.0958t + 1.4876\).

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

Answer

The regression line of \(Y\) on \(X\) is \(u = 0.0958t + 1.4876\).

\(f_{Y|X} (u|t) = \dfrac{2t + 3u^2}{(1 + t)(1 + 4t + t^2)}\) \(0 \le u \le 1 + t\)

\(E[Y|X = t] = \dfrac{1}{(1 + t) (1 + 4t + t^2)} \int_{0}^{1 + t} (2tu + 3u^3)\ du\)

\(= \dfrac{(t + 1)(t + 3) (3t+1)}{4(1 + 4t +t^2)}\), \(0 \le t \le 2\)

tuappr:  [0 2] [0 3] 200 300 (3/88)*(2*t + 3*u.^2).*(u<=1+t)
EYx = sum(u.*P)./sum(P);
eyx = (X+1).*(X+3).*(3*X+1)./(4*(1 + 4*X + X.^2));
plot(X,EYx,X,eyx)            % Plots nearly indistinguishable

Exercise \(\PageIndex{7}\)

(See Exercise 16 from " Problems On Random Vectors and Joint Distributions", Exercise 26 from "Problems on Mathematical Expectation", and Exercise 26 from "Problems on Variance, Covariance, Linear Regression"). \(f_{XY} (t, u) = 12t^2u\) on the parallelogram with vertices

(-1, 0), (0, 0), (1, 1), (0, 1)

The regression line of \(Y\) on \(X\) is \(u = (4t + 5)/9\).

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

Answer

The regression line of \(Y\) on \(X\) is \(u = (23t + 4)/18\).

\(f_{Y|X} (u|t) = I_{[-1, 0]} (t) \dfrac{2u}{(t + 1)^2} + I_{(0, 1]} (t) \dfrac{2u}{(1 - t^2)}\) on the parallelogram

\(E[Y|X = t] = I_{[-1, 0]} (t) \dfrac{1}{(t + 1)^2} \int_{0}^{t + 1} 2u\ du + I_{(0, 1]} (t) \dfrac{1}{(1 - t^2)} \int_{t}^{1} 2u \ du\)

\(= I_{[-1, 0]} (t) \dfrac{2}{3} (t + 1) + I_{(0, 1]} (t) \dfrac{2}{3} \dfrac{t^2 + t + 1}{t + 1}\)

tuappr: [-1 1] [0 1] 200 100 12*t.^2.*u.*((u<= min(t+1,1))&(u>=max(0,t)))
EYx = sum(u.*P)./sum(P);
M = X<=0;
eyx = (2/3)*(X+1).*M + (2/3)*(1-M).*(X.^2 + X + 1)./(X + 1);
plot(X,EYx,X,eyx)            % Plots quite close

Exercise \(\PageIndex{8}\)

(See Exercise 17 from " Problems On Random Vectors and Joint Distributions", Exercise 27 from "Problems on Mathematical Expectation", and Exercise 27 from "Problems on Variance, Covariance, Linear Regression"). \(f_{XY} (t, u) = \dfrac{24}{11} tu\) for \(0 \le t \le 2\), \(0 \le u \le \text{min } \{1, 2 - t\}\).

The regression line of \(Y\) on \(X\) is \(u = (-124t + 368)/431\)

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

Answer

The regression line of \(Y\) on \(X\) is \(u = (-124t + 368)/431\)

\(f_{Y|X} (u|t) = I_{[0, 1]} (t) 2u + I_{(1, 2]} (t) \dfrac{2u}{(2 - t)^2}\)

\(E[Y|X = t] = I_{[0, 1]} (t) \int_{0}^{1} 2u^2 \ du + I_{(1, 2]} (t) \dfrac{1}{(2 - t)^2} \int_{0}^{2 - t} 2u^2 \ du\)

\(= I_{[0, 1]} (t) \dfrac{2}{3} + I_{(1, 2]} (t) \dfrac{2}{3} (2 - t)\)

tuappr: [0 2] [0 1] 200 100 (24/11)*t.*u.*(u<=min(1,2-t))
EYx = sum(u.*P)./sum(P);
M = X <= 1;
eyx = (2/3)*M + (2/3).*(2 - X).*(1-M);
plot(X,EYx,X,eyx)            % Plots quite close

Exercise \(\PageIndex{9}\)

(See Exercise 18 from " Problems On Random Vectors and Joint Distributions", Exercise 28 from "Problems on Mathematical Expectation", and Exercise 28 from "Problems on Variance, Covariance, Linear Regression"). \(f_{XY} (t, u) = \dfrac{3}{23} (t + 2u)\) for \(0 \le t \le 2\), \(0 \le u \le \text{max } \{2 - t, t\}\).

The regression line of \(Y\) on \(X\) is \(u = 1.0561 t - 0.2603\).

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

Answer

The regression line of \(Y\) on \(X\) is \(u = 1.0561 t - 0.2603\).

\(f_{Y|X} (u|t) = I_{[0, 1]} (t) \dfrac{t+2u}{2(2-t)} + I_{(1, 2]} (t) \dfrac{t + 2u}{2t^2}\) \(0 \le u \le \text{max } (2 - t, t)\)

\(E[Y|X = t] = I_{[0, 1]} (t) \dfrac{1}{2(2 - t)} \int_{0}^{2 - t} (tu + 2u^2) \ du + I_{(1, 2]} (t) \dfrac{1}{2t^2} \int_{0}^{t} (tu + 2u^2)\ du\)

\(= I_{[0, 1]} (t) \dfrac{1}{12} (t - 2) ( t - 8) + I_{(1, 2]} (t) \dfrac{7}{12} t\)

tuappr:  [0 2] [0 2] 200 200 (3/23)*(t+2*u).*(u<=max(2-t,t))
EYx = sum(u.*P)./sum(P);
M = X<=1;
eyx = (1/12)*(X-2).*(X-8).*M + (7/12)*X.*(1-M);
plot(X,EYx,X,eyx)             % Plots quite close

Exercise \(\PageIndex{10}\)

(See Exercise 21 from " Problems On Random Vectors and Joint Distributions", Exercise 31 from "Problems on Mathematical Expectation", and Exercise 29 from "Problems on Variance, Covariance, Linear Regression"). \(f_{XY} (t, u) = \dfrac{2}{13} (t + 2u)\) for \(0 \le t \le 2\), \(0 \le u \le \text{min } \{2t, 3 - t\}\).

The regression line of \(Y\) on \(X\) is \(u = -0.1359 t + 1.0839\).

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

Answer

The regression line of \(Y\) on \(X\) is \(u = -0.1359 t + 1.0839\).

\(f_{Y|X} (t|u) = I_{[0, 1]} (t) \dfrac{t + 2u}{6t^2} + I_{(1,2]} (t) \dfrac{t + 2u}{3(3 - t)}\) \(0 \le u \le \text{max } (2t, 3 - t)\)

\(E[Y|X = t] = I_{[0, 1]} (t) \dfrac{1}{6t^2} \int_{0}^{t} (tu + 2u^2)\ du + I_{(1, 2]} (t) \dfrac{1}{3(3 - t)} \int_{0}^{3 - t} (tu + 2u^2)\ du\)

\(= I_{[0, 1]} (t) \dfrac{11}{9} t + I_{(1, 2]} (t) \dfrac{1}{18} (t^2 - 15t + 36)\)

tuappr: [0 2] [0 2] 200 200 (2/13)*(t+2*u).*(u<=min(2*t,3-t))
EYx = sum(u.*P)./sum(P);
M = X<=1;
eyx = (11/9)*X.*M + (1/18)*(X.^2 - 15*X + 36).*(1-M);
plot(X,EYx,X,eyx)              % Plots quite close

Exercise \(\PageIndex{11}\)

(See Exercise 22 from " Problems On Random Vectors and Joint Distributions", Exercise 32 from "Problems on Mathematical Expectation", and Exercise 30 from "Problems on Variance, Covariance, Linear Regression"). \(f_{XY} 9t, u) = I_{[0, 1]} (t) \dfrac{3}{8} (t^2 + 2u) + I_{(1, 2]} (t) \dfrac{9}{14} t^2u^2\). for \(0 \le u \le 1\).

The regression line of \(Y\) on \(X\) is \(u = 0.0817t + 0.5989\).

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

Answer

The regression line of \(Y\) on \(X\) is \(u = 0.0817t + 0.5989\).

\(f_{Y|X} (t|u) = I_{[0, 1]} (t) \dfrac{t^2 + 2u}{t^2 + 1} + I_{(1, 2]} (t) 3u^2\) \(0 \le u \le 1\)

\(E[Y|X = t] = I_{[0, 1]} (t) \dfrac{1}{t^2 + 1} \int_{0}^{1} (t^2u + 2u^2)\ du + I_{(1, 2]} (t) \int_{0}^{1} 3u^3 \ du\)

\(= I_{[0, 1]} (t) \dfrac{3t^2 + 4}{6(t^2 + 1)} + I_{(1, 2]} (t) \dfrac{3}{4}\)

tuappr: [0 2] [0 1] 200 100 (3/8)*(t.^2 + 2*u).*(t<=1) + ...
       (9/14)*t.^2.*u.^2.*(t>1)
EYx = sum(u.*P)./sum(P);
M = X<=1;
eyx = M.*(3*X.^2 + 4)./(6*(X.^2 + 1)) + (3/4)*(1 - M);
plot(X,EYx,X,eyx)              % Plots quite close

For the distributions in Exercises 12-16 below

1. Determine analytically \(E[Z|X = t]\)
2. Use a discrete approximation to calculate the same functions.

Exercise \(\PageIndex{12}\)

\(f_{XY} (t, u) = \dfrac{3}{88} (2t + 3u^2)\) for \(0 \le t \le 2\), \(0 \le u \le 1 + t\), (see Exercise 37 from "Problems on Mathematical Expectation", and Exercise 14.2.6).

\(f_{X} (t) = \dfrac{3}{88} (1 + t) (1 + 4t + t^2) = \dfrac{3}{88} (1 + 5t + 5t^2 + t^3)\), \(0 \le t \le 2\)

\(Z = I_{[0, 1]} (X) 4X + I_{(1, 2]} (X) (X + Y)\)

Answer

\(Z = I_M (X) 4X + I_N (X) (X + Y)\). Use of linearity, (CE8), and (CE10) gives

\(E[Z|X = t] = I_M (t) 4t + I_N(t) (t + E[Y|X = t])\)

\(= I_M (t) 4t + I_N (t) (t + \dfrac{(t + 1)(t + 3) (3t + 1)}{4(1 + 4t + t^2)})\)

% Continuation of Exercise 14.2.6
G = 4*t.*(t<=1) + (t + u).*(t>1);
EZx = sum(G.*P)./sum(P);
M = X<=1;
ezx = 4*X.*M + (X + (X+1).*(X+3).*(3*X+1)./(4*(1 + 4*X + X.^2))).*(1-M);
plot(X,EZx,X,ezx)              % Plots nearly indistinguishable

Exercise \(\PageIndex{13}\)

\(f_{XY} (t, u) = \dfrac{24}{11} tu\) for \(0 \le t \le 2\), \(0 \le u \text{min } \{1, 2 - t\}\) (see Exercise 38 from "Problems on Mathematical Expectaton", Exercise 14.2.8).

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

\(Z = I_{M} (X, Y) \dfrac{1}{2} X + I_M (X, Y) Y^2\), \(M = \{(t ,u): u > t\}\)

Answer

\(Z = I_{M} (X, Y) \dfrac{1}{2} X + I_M (X, Y) Y^2\), \(M = \{(t ,u): u > t\}\)

\(I_M(t, u) = I_{[0, 1]} (t) I_{[t, 1]} (u)\) \(I_{M^c} (t, u) = I_{[0, 1]} (t) I_{[0, t]}(u) + I_{(1, 2]} (t) I_{[0, 2 - t]} (u)\)

\(E[Z|X = t] = I_{[0, 1]} (t) [\dfrac{t}{2} \int_{t}^{1} 2u\ du + \int_{0}^{t} u^2 \cdot 2u\ du] + I_{(1, 2]} (t) \int_{0}^{2 - t} u^2 \cdot \dfrac{2u}{(2 - t)^2}\ du\)

\(= I_{[0, 1]} (t) \dfrac{1}{2} t (1 - t^2 + t^3) + I_{(1, 2]} (t) \dfrac{1}{2} (2- t)^2\)

% Continuation of Exercise 14.2.8
Q = u>t;
G = (1/2)*t.*Q + u.^2.*(1-Q);
EZx = sum(G.*P)./sum(P);
M = X <= 1;
ezx = (1/2)*X.*(1-X.^2+X.^3).*M + (1/2)*(2-X).^2.*(1-M);
plot(X,EZx,X,ezx)              % Plots nearly indistinguishable

Exercise \(\PageIndex{14}\)

\(f_{XY} (t, u) = \dfrac{3}{23} (t + 2u)\) for \(0 \le t \le 2\), \(0 \le u \le \text{max } \{2 - t, t\}\) (see Exercise 39 from "Problems on Mathematical Expectaton", and Exercise 14.2.9).

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

\(Z = I_M (X, Y) (X + Y) + I_{M^c} (X, Y) 2Y\), \(M = \{(t, u): \text{max } (t, u) \le 1\}\)

Answer

\(Z = I_M (X, Y) (X + Y) + I_{M^c} (X, Y) 2Y\), \(M = \{(t, u): \text{max } (t, u) \le 1\}\)

\(I_M (t, u) = I_{[0, 1]} (t) I_{[0, 1]} (u)\) \(I_{M^c} (t, u) = I_{[0, 1]} (t) I_{[1, 2 -t]} (u) + I_{(1,2]} (t) I_{[0, 1]} (u)\)

\(E[Z|X = t] = I_{[0, 1]} (t) \dfrac{1}{2(2 - t)} \int_{0}^{1} (t + u) (t + 2u)\ du + \dfrac{1}{2 - t} \int_{1}^{2 - t} u (t + 2u)\ du] + I_{(1, 2]} (t) 2E [Y|X = t]\)

\(= I_{[0, 1]} (t) \dfrac{1}{12} \cdot \dfrac{2t^3 - 30t^2 + 69t - 60}{t - 2} + I_{(1, 2]} (t) \dfrac{7}{6} 2t\)

% Continuation of Exercise 14.2.9
M = X <= 1;
Q = (t<=1)&(u<=1);
G = (t+u).*Q + 2*u.*(1-Q);
EZx = sum(G.*P)./sum(P);
ezx = (1/12)*M.*(2*X.^3 - 30*X.^2 + 69*X -60)./(X-2) + (7/6)*X.*(1-M);
plot(X,EZx,X,ezx)

Exercise \(\PageIndex{15}\)

\(f_{XY} (t, u) = \dfrac{2}{13} (t + 2u)\), for \(0 \le t \le 2\), \(0 \le u \le \text{min } \{2t, 3 - t\}\). (see Exercise 31 from "Problems on Mathematical Expectaton", and Exercise 14.2.10).

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

\(Z = I_M (X, Y) (X + Y) + I_{M^c} (X, Y) 2Y^2\), \(M = \{(t, u): t \le 1, u \ge 1\}\)

Answer

\(Z = I_M (X, Y) (X + Y) + I_{M^c} (X, Y) 2Y^2\), \(M = \{(t, u): t \le 1, u \ge 1\}\)

\(I_M(t, u) = I_{[0, 1]} (t0 I_{[1, 2]} (u)\) \(I_{M^c} (t, u) = I_{[0, 1]} (t) I_{[0, 1)} (u) + I_{(1, 2]} (t) I_{[0, 3 - t]} (u)\)

\(E[Z|X = t] = I_{[0, 1/2]} (t) \dfrac{1}{6t^2} \int_{0}^{2t} 2u^2 (t + 2u) \ du +\)

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

\(= I_{[0, 1/2]} (t) \dfrac{32}{9} t^2 + I_{(1/2, 1]} (t) \dfrac{1}{36} \cdot \dfrac{80t^3 - 6t^2 - 5t + 2}{t^2} + I_{(1, 2]} (t) \dfrac{1}{9} (- t^3 + 15t^2 - 63t + 81)\)

tuappr:  [0 2] [0 2] 200 200 (2/13)*(t + 2*u).*(u<=min(2*t,3-t))
M = (t<=1)&(u>=1);
Q = (t+u).*M + 2*(1-M).*u.^2;
EZx = sum(Q.*P)./sum(P);
N1 = X <= 1/2;
N2 = (X > 1/2)&(X<=1);
N3 = X > 1;
ezx = (32/9)*N1.*X.^2 + (1/36)*N2.*(80*X.^3 - 6*X.^2 - 5*X + 2)./X.^2 ...
         + (1/9)*N3.*(-X.^3 + 15*X.^2 - 63.*X + 81);
plot(X,EZx,X,ezx)

Exercise \(\PageIndex{16}\)

\(f_{XY} (t, u) = I_{[0, 1]} (t) \dfrac{3}{8} (t^2 + 2u) + I_{(1, 2]} (t) \dfrac{9}{14} t^2 u^2\), for \(0 \le u \le 1\). (see Exercise 32 from "Problems on Mathematical Expectaton", and Exercise 14.2.11).

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

\(Z = I_M (X, Y) X + I_{M^c} (X, Y) XY\), \(M = \{(t, u): u \le \text{min } (1 , 2 - t)\}\)

Answer

\(Z = I_M (X, Y) X + I_{M^c} (X, Y) XY\), \(M = \{(t, u): u \le \text{min } (1 , 2 - t)\}\)

\(E[|X = t] = I_{[0, 1]} (t) \int_{0}^{1} \dfrac{t^3+ 2tu}{t^2 + 1} \ du + I_{(1, 2]} (t) [\int_{0}^{2 - t} 3tu^2\ du + \int_{2 - t}^{1} 3tu^3\ du]\)

\(= I_{[0, 1]} (t) t + I_{(1, 2]} (t) (-\dfrac{13}{4} t+ 12t^2 - 12t^3 + 5t^4 - \dfrac{3}{4} t^5)\)

tuappr:  [0 2] [0 1] 200 100  (t<=1).*(t.^2 + 2*u)./(t.^2 + 1) +3*u.^2.*(t>1)
M = u<=min(1,2-t);
G = M.*t + (1-M).*t.*u;
EZx = sum(G.*P)./sum(P);
N = X<=1;
ezx = X.*N + (1-N).*(-(13/4)*X + 12*X.^2 - 12*X.^3 + 5*X.^4 - (3/4)*X.^5);
plot(X,EZx,X,ezx)

Exercise \(\PageIndex{17}\)

Suppose \(X\) ~ uniform on 0 through \(n\) and \(Y\) ~ conditionally uniform on 0 through \(i\), given \(X = i\).

a. Determine \(E[Y]\) from \(E[Y|X = i]\).
b. Determine the joint distribution for \(\{X, Y\}\) for \(n = 50\) (see Example 7 from "Conditional Expectation, Regression" for a possible approach). Use jcalc to determine \(E[Y]\); compare with the theoretical value.

Answer

a. \(E[Y|X = i] = i/2\), so

\(E[Y] = \sum_{i = 0}^{n} E[Y|X = i] P(X = i) = \dfrac{1}{n + 1} \sum_{i = 1}^{n} i/2 = n/4\)

b. \(P(X = i) = 1/(n + 1)\), \(0 \le i \le n\), \(P(Y = k|X = i) = 1/(i + 1)\). \(0 \le k \le i\); hence
\(P(X = i, Y = k) = 1/(n + 1)(i + 1)\), \(0 \le i \le n\), \(0 \le k \le i\).

n = 50; X = 0:n; Y = 0:n; P0 = zeros(n+1,n+1); for i = 0:n P0(i+1,1:i+1) = (1/((n+1)*(i+1)))*ones(1,i+1); end P = rot90(P0); jcalc: X Y P - - - - - - - - - - - EY = dot(Y,PY) EY = 12.5000 % Comparison with part (a): 50/4 = 12.5

Exercise \(\PageIndex{18}\)

Suppose \(X\) ~ uniform on 1 through \(n\) and \(Y\) ~ conditionally uniform on 1 through \(i\), given \(X = i\).

a. Determine \(E[Y]\) from \(E[Y|X = i]\).
b. Determine the joint distribution for \(\{X, Y\}\) for \(n = 50\) (see Example 7 from "Conditional Expectation, Regression" for a possible approach). Use jcalc to determine \(E[Y]\); compare with the theoretical value.

Answer

a. \(E[Y|X = i] = (i+1)/2\), so

\(E[Y] = \sum_{i = 0}^{n} E[Y|X = i] P(X = i) = \dfrac{1}{n + 1} \sum_{i = 1}^{n} \dfrac{i + 1}{2} = \dfrac{n +3}{4}\)

b. \(P(X = i) = 1/n\), \(1 \le i \le n\), \(P(Y = k|X = i) = 1/i\). \(1 \le k \le i\); hence
\(P(X = i, Y = k) = 1/ni\), \(1 \le i \le n\), \(1 \le k \le i\).

n = 50; X = 1:n; Y = 1:n; P0 = zeros(n,n); for i = 1:n P0(i,1:i) = (1/(n*i))*ones(1,i); end P = rot90(P0); jcalc: P X Y - - - - - - - - - - - - EY = dot(Y,PY) EY = 13.2500 % Comparison with part (a): 53/4 = 13.25

Exercise \(\PageIndex{19}\)

Suppose \(X\) ~ uniform on 1 through \(n\) and \(Y\) ~ conditionally binomial \((i, p)\), given \(X = i\).

a. Determine \(E[Y]\) from \(E[Y|X = k]\).
b. Determine the joint distribution for \(\{X, Y\}\) for \(n = 50\) and \(p = 0.3\). Use jcalc to determine \(E[Y]\); compare with the theoretical value.

Answer

a. \(E[Y|X = i] = ip\), so

\(E[Y] = \sum_{i = 1}^{n} E[Y|X = i] P(X = i) = \dfrac{p}{n} \sum_{i = 1}^{n} i = \dfrac{p(n + 1)}{2}\)

b. \(P(X = i) = 1/n\), \(1 \le i \le n\), \(P(Y = k|X = i)\) = ibinom\((i, p, 0:i)\), \(0 \le k \le i\).

n = 50; p = 0.3; X = 1:n; Y = 0:n; P0 = zeros(n,n+1); % Could use randbern for i = 1:n P0(i,1:i+1) = (1/n)*ibinom(i,p,0:i); end P = rot90(P0); jcalc: X Y P - - - - - - - - - - - EY = dot(Y,PY) EY = 7.6500 % Comparison with part (a): 0.3*51/2 = 0.765

Exercise \(\PageIndex{20}\)

A number \(X\) is selected randomly from the integers 1 through 100. A pair of dice is thrown \(X\) times. Let \(Y\) be the number of sevens thrown on the \(X\) tosses. Determine the joint distribution for \(\{X, Y\}\) and then determine \(E[Y]\).

Answer

a. \(P(X = i) = 1/n\), \(E[Y|X = i] = i/6\), so

\(E[Y] = \dfrac{1}{6} \sum_{i = 0}^{n} i/n = \dfrac{(n + 1)}{12}\)

b. n = 100; p = 1/6; X = 1:n; Y = 0:n; PX = (1/n)*ones(1,n); P0 = zeros(n,n+1); % Could use randbern for i = 1:n P0(i,1:i+1) = (1/n)*ibinom(i,p,0:i); end P = rot90(P0); jcalc EY = dot(Y,PY) EY = 8.4167 % Comparison with part (a): 101/12 = 8.4167

Exercise \(\PageIndex{21}\)

A number \(X\) is selected randomly from the integers 1 through 100. Each of two people draw \(X\) times, independently and randomly, a number from 1 to 10. Let \(Y\) be the number of matches (i.e., both draw ones, both draw twos, etc.). Determine the joint distribution and then determine \(E[Y]\).

Answer

Same as Exercise 14.2.20, except \(p = 1/10\). \(E[Y] = (n + 1)/20\)

n = 100; p = 0.1; X = 1:n; Y = 0:n; PX = (1/n)*ones(1,n);
P0 = zeros(n,n+1);         % Could use randbern
for i = 1:n
  P0(i,1:i+1) = (1/n)*ibinom(i,p,0:i);
end
P = rot90(P0);
jcalc
- - - - - - - - - -
EY = dot(Y,PY)
EY =  5.0500                  % Comparison with part (a): EY = 101/20 = 5.05

Exercise \(\PageIndex{22}\)

\(E[Y|X = t] = 10t\) and \(X\) has density function \(f_X (t) = 4 - 2t\) for \(1 \le t \le 2\). Determine \(E[Y]\).

Answer

\(E[Y] = \int E[Y|X = t] f_X (t)\ dt = \int_{1}^{2} 10t(4 - 2t) \ dt = 40/3\)

Exercise \(\PageIndex{23}\)

\(E[Y|X = t] = \dfrac{2}{3} (1 - t)\) for \(0 \le t < 1\) and \(X\) has density function \(f_X (t) = 30 t^2 ( 1 - t)^2\) for \(0 \le t \le 1\). Determine \(E[Y]\).

Answer

\(E[Y] = \int E[Y|X =t] f_X (t)\ dt = \int_{0}^{1} 20t^2 (1 - t)^3\ dt = 1/3\)

Exercise \(\PageIndex{24}\)

\(E[Y|X = t] = \dfrac{2}{3} (2 - t)\) and \(X\) has density function \(f_X(t) = \dfrac{15}{16} t^2 (2 - t)^2\) \(0 \le t < 2\). Determine \(E[Y]\).

Answer

\(E[Y] = \int E[Y|X =t] f_X(t)\ dt = \dfrac{5}{8} \int_{0}^{2} t^2 (2 - t)^3\ dt = 2/3\)

Exercise \(\PageIndex{25}\)

Suppose the pair \(\{X, Y\}\) is independent, with \(X\) ~ Poisson (\(\mu\)) and \(Y\) ~ Poisson \((\lambda)\). Show that \(X\) is conditionally binomial \((n, \mu/(\mu + \lambda))\), given \(X + Y = n\). That is, show that

\(P(X = k|X + Y = n) = C(n, k) p^k (1 - p)^{n - k}\), \(0 \le k \le n\), for \(p = \mu/(\mu + \lambda)\)

Answer

\(X\) ~ Poisson (\(\mu\)), \(Y\) ~ Poisson \((\lambda)\). Use of property (T1) and generating functions shows that \(X + Y\) ~ Poisson \((\mu + \lambda)\)

\(P(X = k|X + Y = n) = \dfrac{P(X = k, X + Y = n)}{P(X+Y = n)} = \dfrac{P(X = k, Y = n - k)}{P(X + Y) = n}\)

\(= \dfrac{e^{-\mu} \dfrac{\mu^k}{k!} e^{-\lambda} \dfrac{\lambda^{n -k}}{(n - k)!}}{e^{-(\mu + \lambda)} \dfrac{(\mu + \lambda)^n}{n!}} = \dfrac{n!}{k! (n - k)!} \dfrac{\mu^k \lambda^{n - k}}{(\mu + \lambda)^n}\)

Put \(p = \mu/(\mu + \lambda)\) and \(q = 1 - p = \lambda/(\mu + \lambda)\) to get the desired result.

Exercise \(\PageIndex{26}\)

Use the fact that \(g(X, Y) = g^* (X, Y, Z)\), where \(g^* (t, u, v)\) does not vary with \(v\). Extend property (CE10) to show

\(E[g(X, Y)|X = t, Z = v] = E[g(t, Y)|X = t, Z = v]\) a.s. \([P_{XZ}]\)

Answer

\(E[g(X,Y)|X = t, Z = v] = E[g^* (X, Z, Y)| (X, Z) = (t, v)] = E[g^* (t, v, Y)|(X, Z) = (t, v)]\)

\(= E[g(t, Y)|X = t, Z = v]\) a.s. \([P_{XZ}]\) by (CE10)

Exercise \(\PageIndex{27}\)

Use the result of Exercise 14.2.26 and properties (CE9a) and (CE10) to show that

\(E[g(X, Y)|Z = v] = \int E[g(t, Y)|X = t, Z =v] F_{X|Z} (dt|v)\) a.s. \([P_Z]\)

Answer

By (CE9), \(E[g(X, Y)|Z] = E\{E|g(X, Y)|X, Z]|Z\} = E[e(X, Z)|Z]\) a.s.

By (CE10),

\(E[e(X, Z)|Z = v] = E[e(X, v)|Z = v] = \)

\(\int e(t, v) F_{X|Z} (dt|v)\) a.s.

By Exercise 14.2.26,

\(\int E[g(X, Y)|X = t, Z = v] F_{X|Z} (dt|v) = \)

\(\int E[g(t, Y)|X = t, Z = v] F_{X|Z} (dt|v)\) a.s. \([P_Z]\)

Exercise \(\PageIndex{28}\)

A shop which works past closing time to complete jobs on hand tends to speed up service on any job received during the last hour before closing. Suppose the arrival time of a job in hours before closing time is a random variable \(T\) ~ uniform [0, 1]. Service time \(Y\) for a unit received in that period is conditionally exponential \(\beta (2 - u)\), given \(T = u\). Determine the distribution unction for \(Y\).

Answer

\(F_Y (v) = \int F_{Y|T} (v|u) f_T (u)\ du = \int_{0}^{1} (1 - e^{-\beta (2 - u)v})\ du = \)

\(1 - e^{-2\beta v} \dfrac{e^{\beta v} - 1}{\beta v} = 1 - e^{\beta v} [\dfrac{1 - e^{-\beta v}}{\beta v}]\), \(0 < v\)

Exercise \(\PageIndex{29}\)

Time to failure \(X\) of a manufactured unit has an exponential distribution. The parameter is dependent upon the manufacturing process. Suppose the parameter is the value of random variable \(H\) ~ uniform on[0.005, 0.01], and \(X\) is conditionally exponential \((u)\), given \(H = u\). Determine \(P(X > 150)\). Determine \(E[X|H = u]\) and use this to determine \(E[X]\).

Answer

\(F_{X|H} (t|u) = 1 - e^{ut}\) \(f_{H} (u) = \dfrac{1}{0.05} = 200\), \(0.005 \le u \le 0.01\)

\(F_X (t) = 1 - 200 \int_{0.005}^{0.01} e^{-ut}\ du = 1 - \dfrac{200}{t} [e^{-0.005t} - e^{-0.01t}]\)

\(P(X > 150) = \dfrac{200}{150}[e^{-0.75} - e^{-1.5}] \approx 0.3323\)

\(E[X|H = u] = 1/u\) \(E[X] = 200 \int_{0.005}^{0.01} \dfrac{du}{u} = 200 \text{ln } 2\)

Exercise \(\PageIndex{30}\)

A system has \(n\) components. Time to failure of the \(i\)th component is \(X_i\) and the class

\(\{X_i: 1 \le i \le n\}\) is iid exponential (\(\lambda\)). The system fails if any one or more of the components fails. Let \(W\) be the time to system failure. What is the probability the failure is due to the \(i\)th component?

Suggestion. Note that \(W = X_i\) iff \(X_j > X_i\), for all \(j \ne i\). Thus

\(\{W = X_i\} = \{(X_1, X_2, \cdot\cdot\cdot, X_n) \in Q\}\), \(Q = \{(t_1, t_2, \cdot\cdot\cdot t_n): t_k > t_i, \forall k \ne i\}\)

\(P(W = X_i) = E[I_Q (X_1, X_2, \cdot\cdot\cdot, X_n)] = E\{E[I_Q (X_1, X_2, \cdot\cdot\cdot, X_n)|X_i]\}\)

Answer

Let \(Q = \{(t_1, t_2, \cdot\cdot\cdot, t_n): t_k > t_i, k \ne i\}\). Then

\(P(W = X_i) = E[I_Q (X_1, X_2, \cdot\cdot\cdot, X_n)] = E\{E[I_Q (X_1, X_2, \cdot\cdot\cdot, X_n)|X_i]\}\)

\(= \int E[I_Q(X_1, X_2, \cdot\cdot\cdot, t_i, \cdot\cdot\cdot X_n)] F_X (dt)\)

\(E[I_Q (X_1, X_2, \cdot\cdot\cdot, t_i, \cdot\cdot\cdot, X_n)] = \prod_{k \ne i} P(X_k > t) = [1 - F_X (t)]^{n - 1}\)

If \(F_X\) is continuous, strictly increasing, zero for \(t < 0\), put \(u = F_X (t)\), \(du = f_X (t)\ dt\), \(t = 0\) ~ \(u = 0, t = \infty\) ~ \(u = 1\). Then

\(P(W = X_i) = \int_{0}^{1} (1 - u)^{n - 1}\ du = \int_{0}^{1} u^{n - 1}\ du = 1/n\)