(New page: <math>\theta </math> is uniform [0,1] supp. <math>\theta</math> = 1/3 MSE = E[ (<math>\theta</math> - 1/3)^2 ] = E[ <math>\theta</math>^2 - 2*(1/3)*<math>\theta</math> + (1/3)^2 ] = 1/3 ...) |
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| + | [[Category:ECE302Fall2008_ProfSanghavi]] | ||
| + | [[Category:probabilities]] | ||
| + | [[Category:ECE302]] | ||
| + | [[Category:problem solving]] | ||
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<math>\theta </math> is uniform [0,1] | <math>\theta </math> is uniform [0,1] | ||
| − | supp. <math>\theta</math> = 1/3 | + | E[ <math>\theta^2 </math> ] = <math>\int\limits_{0}^{1} \theta^2 d\theta </math> |
| + | = 1/3 | ||
| + | |||
| + | supp. <math>\theta^{hat}</math> = 1/3 | ||
| − | MSE = E[ (<math>\theta</math> - 1/3)^2 ] = E[ <math>\theta</math> | + | MSE = E[ (<math>\theta</math> - 1/3)^2 ] = E[ <math>\theta^2</math> - 2*(1/3)*<math>\theta</math> + (1/3)^2 ] |
= 1/3 - 2(1/3)(1/2) + (1/3)^2 = (1/3)^2 = 1/9 | = 1/3 - 2(1/3)(1/2) + (1/3)^2 = (1/3)^2 = 1/9 | ||
| − | supp. instead <math>\theta</math> = 1/2 | + | supp. instead <math>\theta^{hat}</math> = 1/2 |
| − | MSE = E[ (<math>\theta</math> - 1/2)^2 ] = E[ <math>\theta</math> | + | MSE = E[ (<math>\theta</math> - 1/2)^2 ] = E[ <math>\theta^2</math> - 2*(1/3)*<math>\theta</math> + (1/4) ] = 1/3 - 1/2 + 1/4 = 1/12 |
| + | ---- | ||
| + | [[Main_Page_ECE302Fall2008sanghavi|Back to ECE302 Fall 2008 Prof. Sanghavi]] | ||
Latest revision as of 12:41, 22 November 2011
$ \theta $ is uniform [0,1]
E[ $ \theta^2 $ ] = $ \int\limits_{0}^{1} \theta^2 d\theta $ = 1/3
supp. $ \theta^{hat} $ = 1/3
MSE = E[ ($ \theta $ - 1/3)^2 ] = E[ $ \theta^2 $ - 2*(1/3)*$ \theta $ + (1/3)^2 ] = 1/3 - 2(1/3)(1/2) + (1/3)^2 = (1/3)^2 = 1/9
supp. instead $ \theta^{hat} $ = 1/2
MSE = E[ ($ \theta $ - 1/2)^2 ] = E[ $ \theta^2 $ - 2*(1/3)*$ \theta $ + (1/4) ] = 1/3 - 1/2 + 1/4 = 1/12
