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I have a random sample of size 3 denoted by X below and it comes from a normal distribution with mean 7 and variance 14. I have the matrix A shown below. I am looking for E[Q]. I know that E[Q] = 1/sigma^2 * E[Q]. The formula from the textbook for E[Q] is shown below where sigma is the variance-covariance matrix.

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I am having trouble with the following two items:

  1. What is sigma? I am unsure how to find the variance-covariance matrix. Is it simply just a 3x3 matrix with 14 on the diagonals?

  2. What is $\mu$? Is it [7 7 7]?

Thanks for the help.

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    Since sigma^2 is known, then can't we treat it as a constant, and take it out of the expectation? Also, is the solution below correct for E[Q] or is it just to find sigma?2012-03-19

1 Answers 1

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Write $E[Q]$ as $E[Q] = E\left[\frac{1}{\sigma^2}\sum_{i=1}^3\sum_{j=1}^3 a_{i,j}X_iX_j\right] = \frac{1}{\sigma^2}\sum_{i=1}^3\sum_{j=1}^3 a_{i,j}E[X_iX_j],$ replace $E[X_iX_j]$ by $\text{cov}(X_i,X_j) + E[X_i]E[X_j] = \begin{cases} \mu_i\mu_j, &i \neq j,\\\sigma^2 + \mu_i^2, & i = j,\end{cases}$ and rearrange.

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    Just to confirm, this is to find the variance-covariance matrix?2012-03-19