Finding the Expected Value of T=(sumX_i^2, sum X_i)

tramolatzqvg

tramolatzqvg

Answered question

2022-11-03

Finding the Expected Value of T = X i 2

Answer & Explanation

Lillianna Salazar

Lillianna Salazar

Beginner2022-11-04Added 22 answers

The given density is not a general Gaussian Family but it is a particular one of Gaussian, Centered in ZERO. Thus the solution is very simple
(a) A sufficient statistic is T = i = 1 n X i 2
In fact the likelihood is the following
L ( θ ) = ( 1 2 π θ ) n 2 e 1 2 θ i X i 2
(b)
E [ T ] = E [ i = 1 n X i 2 ] = n E [ X 1 2 ] = n θ
(c)
T = T n

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