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.

  • $\begingroup$ By "it comes from" do you mean each component is an iid thereof? $\endgroup$ – anon Mar 19 '12 at 4:15
  • $\begingroup$ All the problem says is that Let X' = [X1, X2, X3] denote a random sample of size 3 from the normal distribution I specified above. I guess since it doesn't say otherwise, I assume they are iid. $\endgroup$ – icobes Mar 19 '12 at 4:48
  • $\begingroup$ @icobes Your formula that begins "I know that..." doesn't seem right. $\endgroup$ – user940 Mar 19 '12 at 15:39
  • $\begingroup$ 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? $\endgroup$ – icobes Mar 19 '12 at 20:18

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.

  • $\begingroup$ Is this to find E[Q]? Or to find sigma? Would all the non-diagonal entries be 49 according to this? $\endgroup$ – icobes Mar 19 '12 at 20:19
  • $\begingroup$ Try re-arranging as I suggested and see if you get the result that you state is the answer in generic terms without use of any numerical values. $\endgroup$ – Dilip Sarwate Mar 19 '12 at 20:30
  • $\begingroup$ Just to confirm, this is to find the variance-covariance matrix? $\endgroup$ – icobes Mar 19 '12 at 20:31

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