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 
 
 
TABLE OF CONTENTS 
 
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 
 
 
TABLE OF CONTENTS 
 
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 
 
 
TABLE OF CONTENTS 
 
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 
 
 
TABLE OF CONTENTS 
 
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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