12.3 Problems on variance, covariance, linear regression (Page 3/5)

Applied probability Page 3 / 5

For the Beta $(r, s)$ distribution,

Determine $E [X^{n}]$ , where n is a positive integer.
Use the result of part (a) to determine $E [X]$ and $Var [X]$ .

E [X^{n}] = \frac{Γ (r + s)}{Γ (r) Γ (s)} \int_{0}^{1} t^{r + n - 1} {(1 - t)}^{s - 1} d t = \frac{Γ (r + s)}{Γ (r) Γ (s)} \cdot \frac{Γ (r + n) Γ (s)}{Γ (r + s + n)} =

\frac{Γ (r + n) Γ (r + s)}{Γ (r + s + n) Γ (r)}

Using $Γ (x + 1) = x Γ (x)$ we have

E [X] = \frac{r}{r + s}, E [X^{2}] = \frac{r (r + 1)}{(r + s) (r + s + 1)}

Some algebraic manipulations show that

Var [X] = E [X^{2}] - E^{2} [X] = \frac{r s}{{(r + s)}^{2} (r + s + 1)}

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The pair ${X, Y}$ has joint distribution. Suppose

E [X] = 3, E [X^{2}] = 11, E [Y] = 10, E [Y^{2}] = 101, E [X Y] = 30

Determine $Var [15 X - 2 Y]$ .

EX = 3;
EX2 = 11;EY = 10;
EY2 = 101;EXY = 30;
VX = EX2 - EX^2VX =    2
VY = EY2 - EY^2VY =    1
CV = EXY - EX*EYCV =    0
VZ = 15^2*VX + 2^2*VYVZ =  454

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The pair ${X, Y}$ has joint distribution. Suppose

E [X] = 2, E [X^{2}] = 5, E [Y] = 1, E [Y^{2}] = 2, E [X Y] = 1

Determine $Var [3 X + 2 Y]$ .

EX = 2;
EX2 = 5;EY = 1;
EY2 = 2;EXY = 1;
VX = EX2 - EX^2VX =    1
VY = EY2 - EY^2VY =    1
CV = EXY - EX*EYCV =   -1
VZ = 9*VX + 4*VY + 2*6*CVVZ =    1

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The pair ${X, Y}$ is independent, with

E [X] = 2, E [Y] = 1, Var [X] = 6, Var [Y] = 4

Let $Z = 2 X^{2} + X Y^{2} - 3 Y + 4.$ .
Determine $E [Z].$

EX = 2;
EY = 1;VX = 6;
VY = 4;EX2 = VX + EX^2
EX2 =  10EY2 = VY + EY^2
EY2 =   5EZ = 2*EX2 + EX*EY2 - 3*EY + 4
EZ =   31

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(See Exercise 9 from "Problems on Mathematical Expectation"). Random variable X has density function

f_{X} (t) = \{\begin{matrix} (6 / 5) t^{2} & for 0 \leq t \leq 1 \\ (6 / 5) (2 - t) & for 1 < t \leq 2 \end{matrix}) = I [0, 1] (t) \frac{6}{5} t^{2} + I_{(1, 2]} (t) \frac{6}{5} (2 - t)

$E [X] = 11 / 10$ . Determine $Var [X]$ .

E [X^{2}] = \int t^{2} f_{X} (t) d t = \frac{6}{5} \int_{0}^{1} t^{4} d t + \frac{6}{5} \int_{1}^{2} (2 t^{2} - t^{3}) d t = \frac{67}{50}

Var [X] = E [X^{2}] - E^{2} [X] = \frac{13}{100}

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For the distributions in Exercises 20-22

Determine $Var [X]$ , $Cov [X, Y]$ , and the regression line of Y on X .

(See Exercise 7 from "Problems On Random Vectors and Joint Distributions", and 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


npr08_07 Data are in X, Y, P
jcalc- - - - - - - - - - -
EX = dot(X,PX);EY = dot(Y,PY);
VX = dot(X.^2,PX) - EX^2VX =   5.1116
CV = total(t.*u.*P) - EX*EYCV =   2.6963
a = CV/VXa =    0.5275
b = EY - a*EXb =    0.6924       % Regression line: u = at + b

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(See Exercise 8 from "Problems On Random Vectors and Joint Distributions", and 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


npr08_08 Data are in X, Y, P
jcalc- - - - - - - - - - - -
EX = dot(X,PX);EY = dot(Y,PY);
VX = dot(X.^2,PX) - EX^2VX =  31.0700
CV = total(t.*u.*P) - EX*EYCV =  -8.0272
a  = CV/VXa  =  -0.2584
b = EY - a*EXb  =   5.6110       % Regression line: u = at + b

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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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