Presentation is loading. Please wait.

Presentation is loading. Please wait.

Definition of, the expected value of a function of X : 1 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE To find the expected value of a function of.

Published byMae Wilkinson Modified over 8 years ago

Similar presentations

Presentation on theme: "Definition of, the expected value of a function of X : 1 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE To find the expected value of a function of."— Presentation transcript:

1 Definition of, the expected value of a function of X : 1 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE To find the expected value of a function of a random variable, one calculates all the possible values of the function, weights them by the corresponding probabilities, and sums the results.

2 Definition of, the expected value of a function of X : Example: For example, the expected value of X 2 is found by calculating all its possible values, multiplying them by the corresponding probabilities, and summing. 2 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE

3 3 x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 The calculation of the expected value of a function of a random variable will be outlined in general and then illustrated with an example.

4 4 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 First one makes a list of the possible values of X and the corresponding probabilities.

5 5 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 Next the function of X is calculated for each possible value of X.

6 6 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 Then, one at a time, the value of the function is weighted by its corresponding probability.

7 7 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 This is done individually for each possible value of X.

8 8 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 The sum of the weighted values is the expected value of the function of X.

9 9 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 The process will be illustrated for X 2, where X is the random variable defined in the first sequence. The 11 possible values of X and the corresponding probabilities are listed.

10 10 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 First one calculates the possible values of X 2.

11 11 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 The first value is 4, which arises when X is equal to 2. The probability of X being equal to 2 is 1/36, so the weighted function is 4/36, which we shall write in decimal form as 0.11.

12 12 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 Similarly for all the other possible values of X.

13 13 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE The expected value of X 2 is the sum of its weighted values in the final column. It is equal to 54.83. It is the average value of the figures in the previous column, taking the differing probabilities into account. x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83

14 x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 Note that E(X 2 ) is not the same thing as E(X), squared. In the previous sequence we saw that E(X) for this example was 7. Its square is 49. 14 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE

15 Copyright Christopher Dougherty 2012. These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section R.2 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/http://www.oup.com/uk/orc/bin/9780199567089/. Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course EC2020 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 2012.10.29

Download ppt "Definition of, the expected value of a function of X : 1 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE To find the expected value of a function of."

Similar presentations

EXPECTED VALUE RULES 1. This sequence states the rules for manipulating expected values. First, the additive rule. The expected value of the sum of two.

EXPECTED VALUE RULES 1. This sequence states the rules for manipulating expected values. First, the additive rule. The expected value of the sum of two.
THE ERROR CORRECTION MODEL 1 The error correction model is a variant of the partial adjustment model. As with the partial adjustment model, we assume a.

THE ERROR CORRECTION MODEL 1 The error correction model is a variant of the partial adjustment model. As with the partial adjustment model, we assume a.
To find the expected value of a function of a random variable, you calculate all the possible values of the function, weight them by the corresponding.

To find the expected value of a function of a random variable, you calculate all the possible values of the function, weight them by the corresponding.
Definition of, the expected value of a function of X : 1 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE To find the expected value of a function of.

Definition of, the expected value of a function of X : 1 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE To find the expected value of a function of.
1 This very short sequence presents an important definition, that of the independence of two random variables. Two random variables X and Y are said to.

1 This very short sequence presents an important definition, that of the independence of two random variables. Two random variables X and Y are said to.
Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: expected value of a random variable Original citation: Dougherty,

Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: expected value of a random variable Original citation: Dougherty,
The third sequence defined the expected value of a function of a random variable X. There is only one function that is of much interest to us, at least.

The third sequence defined the expected value of a function of a random variable X. There is only one function that is of much interest to us, at least.
1 THE DISTURBANCE TERM IN LOGARITHMIC MODELS Thus far, nothing has been said about the disturbance term in nonlinear regression models.

1 THE DISTURBANCE TERM IN LOGARITHMIC MODELS Thus far, nothing has been said about the disturbance term in nonlinear regression models.
1 XX X1X1 XX X Random variable X with unknown population mean  X function of X probability density Sample of n observations X 1, X 2,..., X n : potential.

1 XX X1X1 XX X Random variable X with unknown population mean  X function of X probability density Sample of n observations X 1, X 2,..., X n : potential.
Christopher Dougherty EC220 - Introduction to econometrics (chapter 12) Slideshow: dynamic model specification Original citation: Dougherty, C. (2012)

Christopher Dougherty EC220 - Introduction to econometrics (chapter 12) Slideshow: dynamic model specification Original citation: Dougherty, C. (2012)
1 THE NORMAL DISTRIBUTION In the analysis so far, we have discussed the mean and the variance of a distribution of a random variable, but we have not said.

1 THE NORMAL DISTRIBUTION In the analysis so far, we have discussed the mean and the variance of a distribution of a random variable, but we have not said.
1 BINARY CHOICE MODELS: PROBIT ANALYSIS In the case of probit analysis, the sigmoid function F(Z) giving the probability is the cumulative standardized.

1 BINARY CHOICE MODELS: PROBIT ANALYSIS In the case of probit analysis, the sigmoid function F(Z) giving the probability is the cumulative standardized.
1 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red123456 This sequence provides an example of a discrete random variable. Suppose that you.

1 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red This sequence provides an example of a discrete random variable. Suppose that you.
Random effects estimation RANDOM EFFECTS REGRESSIONS When the observed variables of interest are constant for each individual, a fixed effects regression.

Random effects estimation RANDOM EFFECTS REGRESSIONS When the observed variables of interest are constant for each individual, a fixed effects regression.
1 ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA Assumptions C.1, C.3, C.4, C.5, and C.8, and the consequences of their violations are the.

1 ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA Assumptions C.1, C.3, C.4, C.5, and C.8, and the consequences of their violations are the.
00  0 +1.96sd  0 –sd  0 –1.96sd  0 +sd 2.5% CONFIDENCE INTERVALS probability density function of X null hypothesis H 0 :  =  0 In the sequence.

00  sd  0 –sd  0 –1.96sd  0 +sd 2.5% CONFIDENCE INTERVALS probability density function of X null hypothesis H 0 :  =  0 In the sequence.
EXPECTED VALUE OF A RANDOM VARIABLE 1 The expected value of a random variable, also known as its population mean, is the weighted average of its possible.

EXPECTED VALUE OF A RANDOM VARIABLE 1 The expected value of a random variable, also known as its population mean, is the weighted average of its possible.
Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: expected value of a function of a random variable Original citation:

Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: expected value of a function of a random variable Original citation:
Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: confidence intervals Original citation: Dougherty, C. (2012) EC220.

Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: confidence intervals Original citation: Dougherty, C. (2012) EC220.
Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: continuous random variables Original citation: Dougherty, C. (2012)

Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: continuous random variables Original citation: Dougherty, C. (2012)

Similar presentations

Ads by Google