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- Applied probability
- Variance, covariance, linear
- Problems on variance, covariance
For the Beta
distribution,
- Determine
, where
n is a positive integer.
- Use the result of part (a) to determine
and
.
Using
we have
Some algebraic manipulations show that
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The pair
has joint distribution. Suppose
Determine
.
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
has joint distribution. Suppose
Determine
.
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
is independent, with
Let
.
Determine
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
. Determine
.
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For the distributions in Exercises 20-22
Determine
,
, 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
has the joint distribution (in
file
npr08_07.m ):
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
has the joint distribution (in
file
npr08_08.m ):
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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