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# Mathematical expectation continuous,STATISTICAL METHODS IN ECONOMICS-1 Economics Notes | EduRev

## Economics : Mathematical expectation continuous,STATISTICAL METHODS IN ECONOMICS-1 Economics Notes | EduRev

``` Page 1

Mathematical expectation continuous
Institute of Lifelong Learning, University of Delhi 1

DC-1
Semester-II
Paper-III: Statistical Methods in Economics-I
Lesson: Mathematical expectation continuous
Lesson Developer: Chandra Goswami
College/Department: Department of Economics,
Dyal Singh College, University of Delhi

Page 2

Mathematical expectation continuous
Institute of Lifelong Learning, University of Delhi 1

DC-1
Semester-II
Paper-III: Statistical Methods in Economics-I
Lesson: Mathematical expectation continuous
Lesson Developer: Chandra Goswami
College/Department: Department of Economics,
Dyal Singh College, University of Delhi

Mathematical expectation continuous
Institute of Lifelong Learning, University of Delhi 2

Section Number and Heading            Page Number
Learning Objectives             2
1.   Expected value of a continuous random variable                              2
2.   Expectation of a function of a continuous random variable       4
3.  Variance of a continuous random variable         8
4.  Variance of a function of a continuous random variable                           10
5.  Rules of mathematical expectation                   12
6.  Expectation and variance of sums of continuous random variables                 14
7.  Characteristics of the probability density function      16
Practice Questions                      18

Content Developer
Chandra Goswami, Associate Professor, Department of Economics
Dyal Singh College, University of Delhi

Reference
Jay L. Devore: Probability and Statistics for Engineering and the Sciences,
Cengage Learning, 8
th
edition [Chapter 4]

MATHEMATICAL EXPECTATION: CONTINUOUS RANDOM VARIABLES

Learning objectives:
Page 3

Mathematical expectation continuous
Institute of Lifelong Learning, University of Delhi 1

DC-1
Semester-II
Paper-III: Statistical Methods in Economics-I
Lesson: Mathematical expectation continuous
Lesson Developer: Chandra Goswami
College/Department: Department of Economics,
Dyal Singh College, University of Delhi

Mathematical expectation continuous
Institute of Lifelong Learning, University of Delhi 2

Section Number and Heading            Page Number
Learning Objectives             2
1.   Expected value of a continuous random variable                              2
2.   Expectation of a function of a continuous random variable       4
3.  Variance of a continuous random variable         8
4.  Variance of a function of a continuous random variable                           10
5.  Rules of mathematical expectation                   12
6.  Expectation and variance of sums of continuous random variables                 14
7.  Characteristics of the probability density function      16
Practice Questions                      18

Content Developer
Chandra Goswami, Associate Professor, Department of Economics
Dyal Singh College, University of Delhi

Reference
Jay L. Devore: Probability and Statistics for Engineering and the Sciences,
Cengage Learning, 8
th
edition [Chapter 4]

MATHEMATICAL EXPECTATION: CONTINUOUS RANDOM VARIABLES

Learning objectives:
Mathematical expectation continuous
Institute of Lifelong Learning, University of Delhi 3
In this chapter you will learn how to obtain two main characteristics of the
probability distribution of a continuous random variable. You will learn how to derive
the mean and variance of distributions of continuous random variables. You will also
learn how to apply the rules of mathematical expectation to functions of random
variables as well as to sums of random variables. The mean, variance, median and
mode, and coefficients of skewness and kurtosis will help you to identify the
characteristics and shape of the distribution.

Chapter Outline
1. Expected value of a continuous random variable
2. Expectation of a function of a continuous random variable
3. Variance of a continuous random variable
4. Variance of a function of a continuous random variable
5. Rules of mathematical expectation
6. Expectation and variance of sums of continuous random variables
7. Characteristics of the probability density function

1  EXPECTED VALUE OF A CONTINUOUS RANDOM VARIABLE
The mean of a distribution is the point on the number line where the distribution is
centered. The mean of the distribution of a continuous random variable (probability
density function or pdf) is its expected value. Expected value of a continuous random
variable is obtained as a weighted average of the values of the rv where the
probability densities are the weights. For discrete random variables method of
summation was used. In case of continuous random variables, expected value is
obtained by method of integration.

Definition 1
The expected value or mean value of a continuous random variable X with probability
density function f(x) is
Page 4

Mathematical expectation continuous
Institute of Lifelong Learning, University of Delhi 1

DC-1
Semester-II
Paper-III: Statistical Methods in Economics-I
Lesson: Mathematical expectation continuous
Lesson Developer: Chandra Goswami
College/Department: Department of Economics,
Dyal Singh College, University of Delhi

Mathematical expectation continuous
Institute of Lifelong Learning, University of Delhi 2

Section Number and Heading            Page Number
Learning Objectives             2
1.   Expected value of a continuous random variable                              2
2.   Expectation of a function of a continuous random variable       4
3.  Variance of a continuous random variable         8
4.  Variance of a function of a continuous random variable                           10
5.  Rules of mathematical expectation                   12
6.  Expectation and variance of sums of continuous random variables                 14
7.  Characteristics of the probability density function      16
Practice Questions                      18

Content Developer
Chandra Goswami, Associate Professor, Department of Economics
Dyal Singh College, University of Delhi

Reference
Jay L. Devore: Probability and Statistics for Engineering and the Sciences,
Cengage Learning, 8
th
edition [Chapter 4]

MATHEMATICAL EXPECTATION: CONTINUOUS RANDOM VARIABLES

Learning objectives:
Mathematical expectation continuous
Institute of Lifelong Learning, University of Delhi 3
In this chapter you will learn how to obtain two main characteristics of the
probability distribution of a continuous random variable. You will learn how to derive
the mean and variance of distributions of continuous random variables. You will also
learn how to apply the rules of mathematical expectation to functions of random
variables as well as to sums of random variables. The mean, variance, median and
mode, and coefficients of skewness and kurtosis will help you to identify the
characteristics and shape of the distribution.

Chapter Outline
1. Expected value of a continuous random variable
2. Expectation of a function of a continuous random variable
3. Variance of a continuous random variable
4. Variance of a function of a continuous random variable
5. Rules of mathematical expectation
6. Expectation and variance of sums of continuous random variables
7. Characteristics of the probability density function

1  EXPECTED VALUE OF A CONTINUOUS RANDOM VARIABLE
The mean of a distribution is the point on the number line where the distribution is
centered. The mean of the distribution of a continuous random variable (probability
density function or pdf) is its expected value. Expected value of a continuous random
variable is obtained as a weighted average of the values of the rv where the
probability densities are the weights. For discrete random variables method of
summation was used. In case of continuous random variables, expected value is
obtained by method of integration.

Definition 1
The expected value or mean value of a continuous random variable X with probability
density function f(x) is
Mathematical expectation continuous
Institute of Lifelong Learning, University of Delhi 4
?
?
? ?
? ? dx x f x X E
X
) ( . ) ( ?

When the pdf f(x) specifies a model for the distribution of X values in a numerical
population, then µ
X
is the population mean.

Example 1.1
If a contractor’s profits on a construction job can be looked upon as a continuous rv
having the pdf

?
?
?
?
?
? ? ? ?
?
otherwise
x x
x f
0
5 1 ) 1 (
18
1
) (
where the units are \$1000, her expected profit is
dx x x
dx x x X E
) (
18
1
) 1 (
18
1
. ) (
5
1
2
? ?
? ?
?
?
?
?
? ?

?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
? ?
?
2
1
3
1
2
25
3
125
18
1
2 3 18
1
5
1
2 3
x x

?
?
?
?
?
?
? ?
2
24
3
126
18
1

3
18
54
18
12 42
? ?
?
?
Therefore, expected profit is \$3,000

Exercise 1
The tread wear (in thousands of kilometers) that car owners get with a certain kind of
tyre is a rv X whose pdf is given by

?
?
?
?
?
?
?
?
?
?
?
0 0
0
30
1
) (
30
x
x e
x f
x

Page 5

Mathematical expectation continuous
Institute of Lifelong Learning, University of Delhi 1

DC-1
Semester-II
Paper-III: Statistical Methods in Economics-I
Lesson: Mathematical expectation continuous
Lesson Developer: Chandra Goswami
College/Department: Department of Economics,
Dyal Singh College, University of Delhi

Mathematical expectation continuous
Institute of Lifelong Learning, University of Delhi 2

Section Number and Heading            Page Number
Learning Objectives             2
1.   Expected value of a continuous random variable                              2
2.   Expectation of a function of a continuous random variable       4
3.  Variance of a continuous random variable         8
4.  Variance of a function of a continuous random variable                           10
5.  Rules of mathematical expectation                   12
6.  Expectation and variance of sums of continuous random variables                 14
7.  Characteristics of the probability density function      16
Practice Questions                      18

Content Developer
Chandra Goswami, Associate Professor, Department of Economics
Dyal Singh College, University of Delhi

Reference
Jay L. Devore: Probability and Statistics for Engineering and the Sciences,
Cengage Learning, 8
th
edition [Chapter 4]

MATHEMATICAL EXPECTATION: CONTINUOUS RANDOM VARIABLES

Learning objectives:
Mathematical expectation continuous
Institute of Lifelong Learning, University of Delhi 3
In this chapter you will learn how to obtain two main characteristics of the
probability distribution of a continuous random variable. You will learn how to derive
the mean and variance of distributions of continuous random variables. You will also
learn how to apply the rules of mathematical expectation to functions of random
variables as well as to sums of random variables. The mean, variance, median and
mode, and coefficients of skewness and kurtosis will help you to identify the
characteristics and shape of the distribution.

Chapter Outline
1. Expected value of a continuous random variable
2. Expectation of a function of a continuous random variable
3. Variance of a continuous random variable
4. Variance of a function of a continuous random variable
5. Rules of mathematical expectation
6. Expectation and variance of sums of continuous random variables
7. Characteristics of the probability density function

1  EXPECTED VALUE OF A CONTINUOUS RANDOM VARIABLE
The mean of a distribution is the point on the number line where the distribution is
centered. The mean of the distribution of a continuous random variable (probability
density function or pdf) is its expected value. Expected value of a continuous random
variable is obtained as a weighted average of the values of the rv where the
probability densities are the weights. For discrete random variables method of
summation was used. In case of continuous random variables, expected value is
obtained by method of integration.

Definition 1
The expected value or mean value of a continuous random variable X with probability
density function f(x) is
Mathematical expectation continuous
Institute of Lifelong Learning, University of Delhi 4
?
?
? ?
? ? dx x f x X E
X
) ( . ) ( ?

When the pdf f(x) specifies a model for the distribution of X values in a numerical
population, then µ
X
is the population mean.

Example 1.1
If a contractor’s profits on a construction job can be looked upon as a continuous rv
having the pdf

?
?
?
?
?
? ? ? ?
?
otherwise
x x
x f
0
5 1 ) 1 (
18
1
) (
where the units are \$1000, her expected profit is
dx x x
dx x x X E
) (
18
1
) 1 (
18
1
. ) (
5
1
2
? ?
? ?
?
?
?
?
? ?

?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
? ?
?
2
1
3
1
2
25
3
125
18
1
2 3 18
1
5
1
2 3
x x

?
?
?
?
?
?
? ?
2
24
3
126
18
1

3
18
54
18
12 42
? ?
?
?
Therefore, expected profit is \$3,000

Exercise 1
The tread wear (in thousands of kilometers) that car owners get with a certain kind of
tyre is a rv X whose pdf is given by

?
?
?
?
?
?
?
?
?
?
?
0 0
0
30
1
) (
30
x
x e
x f
x

Mathematical expectation continuous
Institute of Lifelong Learning, University of Delhi 5
What tread wear can a car owner expect to get with one of the tyres?
Solution
dx e x
dx e x X E
x
x
?
?
?
?
?
?
? ?
?
?
0
30
30
.
30
1
30
1
. ) (

Integrating by parts,
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
? ? ?
? ?
?
?
?
0 0
30
0
30
1
30
1
) ( dx e dx e x X E
x x

?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
0 30
1
30
0
30
1
30
30
1
dx
e e
x
x x

? ?
30
1
0
0
30
1
30
1
0
30
1
30
0
30
?
?
?
?
?
?
?
?
?
? ? ?
? ? ? ?
?
?
?
?
?
x
x
e
dx e

Therefore, average tread wear a car owner can expect to get is 30,000 km.

2 EXPECTED VALUE OF A FUNCTION OF A CONTINUOUS RANDOM
VARIABLE
If X is a continuous rv with probability density function f(x), then any function of X,
h(X), will also have the pdf f(x).

Definition 2
If X is a continuous random variable with pdf f(x) and h(X) is any function of X, then
?
?
? ?
? ? dx x f x h X h E
X h
) ( ) ( )] ( [
) (
?
provided that
?
?
? ?
? ? dx x f x h ) ( ) (

Proposition 1
If h(X) is a linear function such as h(X) = aX + b, then E[h(X)] = aE(X) + b
Proof:
```
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