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Use the table above to calculate the expected value.

Use the table above to calculate the expected value.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Suppose you rolled a fair six-sided die a few times and recorded the results. (You would have a collection of numbers 1-6.) Now, suppose you wanted to average these numbers. What value would you expect? If the die was rolled 6 times and the results were: 1, 4, 2, 2, 3, 2, then the average of this sample of rolls would be 2.333. This experiment could be peformed again: 6, 5, 2, 1, 1, 4. Now, the average of this sample is 3.167. Obviously, these small samples have a lot of variation in them; however, is there a way to calculate the expected average? If we were to construct a very large sample, rolling this die 1 million times, what would the expected value (mean) be? Luckily, the rules of probability allows us to do this calculation without actually having to roll the die 1 million times! Below, you will find a probability table where all of the possible outcomes from the die are listed. The first column contains the possible outcomes on the die, the second column contains the probability of each outcome (since the die is fair, all outcomes are equally likely), the remaining columns are values used to calculate the expected value and variance. Values Probability ХP(x) (x - ) - P(x) (x) P(x) 1 - = 0.16 1×0.16= 0.15=0.167(1-3.5)². 1.042 1 1 = 0.16 2 x 0.16 = 0.3=0.33 (2 - 3.5)2. =0.375 6. 0.16 3x0.16 = 0.5 (3– 3.5)2. =0.042 = 0.16 4x 0.16 = 0.6= 0.67 (4- 3.5)2. 4 20.042 =0.16 20.375 5x 0.16 =0.16 0.83 (5- 3.5)2. (6 – 3.5)2 - () =0.16 6x 0.16 = 1 =1.042 Expected Value (Mean): Ux=??? Variance: o2 = E (x;- ²· P(x;)=2.918 (Sum of 4th column) i=1 6 2(x;- H)<. P(x;) - /2.918 1.708 (Square root of the variance) j =1 Standard Deviation: %3D Use the table above to calculate the expected value. 비6 16 CO
Transcribed Image Text:Suppose you rolled a fair six-sided die a few times and recorded the results. (You would have a collection of numbers 1-6.) Now, suppose you wanted to average these numbers. What value would you expect? If the die was rolled 6 times and the results were: 1, 4, 2, 2, 3, 2, then the average of this sample of rolls would be 2.333. This experiment could be peformed again: 6, 5, 2, 1, 1, 4. Now, the average of this sample is 3.167. Obviously, these small samples have a lot of variation in them; however, is there a way to calculate the expected average? If we were to construct a very large sample, rolling this die 1 million times, what would the expected value (mean) be? Luckily, the rules of probability allows us to do this calculation without actually having to roll the die 1 million times! Below, you will find a probability table where all of the possible outcomes from the die are listed. The first column contains the possible outcomes on the die, the second column contains the probability of each outcome (since the die is fair, all outcomes are equally likely), the remaining columns are values used to calculate the expected value and variance. Values Probability ХP(x) (x - ) - P(x) (x) P(x) 1 - = 0.16 1×0.16= 0.15=0.167(1-3.5)². 1.042 1 1 = 0.16 2 x 0.16 = 0.3=0.33 (2 - 3.5)2. =0.375 6. 0.16 3x0.16 = 0.5 (3– 3.5)2. =0.042 = 0.16 4x 0.16 = 0.6= 0.67 (4- 3.5)2. 4 20.042 =0.16 20.375 5x 0.16 =0.16 0.83 (5- 3.5)2. (6 – 3.5)2 - () =0.16 6x 0.16 = 1 =1.042 Expected Value (Mean): Ux=??? Variance: o2 = E (x;- ²· P(x;)=2.918 (Sum of 4th column) i=1 6 2(x;- H)<. P(x;) - /2.918 1.708 (Square root of the variance) j =1 Standard Deviation: %3D Use the table above to calculate the expected value. 비6 16 CO
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