of . The matrix equation can be solved by well known methods such as Gauss elimination method. A shorter, non-numerical example can be found in orthogonality principle. Example 2 Consider a vector formed by taking observations of a fixed but unknown scalar parameter disturbed by white Gaussian noise. We can describe the process by a linear equation , where . Depending on context it will be clear if represents a scalar or a vector. Suppose that we know to be the range within which the value of is going to fall in. We can model our uncertainty of by an aprior uniform distribution over an interval , and thus will have variance of . Let the noise vector be normally distributed as where is an identity matrix. Also and are independent and . It is easy to see that Thus, the linear MMSE estimator is given by We can simplify the expression by using the alternative form for as where for we have Similarly, the variance of the estimator is Thus the MMSE of this linear estimator is For very large , we see that the MMSE estimator of a scalar with uniform aprior distribution can be approximated by the arithmetic average of all the observed data while the variance will be unaffected by data and the LMMSE of the estimate will tend to zero. However, the estimator is suboptimal since it is constrained to be linear. Had the random variable also been Gaussian, then the estimator would have been optimal. Notice, that the form of the estimator will remain unchanged, regardless of the apriori distribution of , so long as the mean and variance of these distributions are the same. Example 3 Consider a variation of the above example: Two candidates are standing for an election. Let the fraction of votes that a candidate will receive on an election day be Thus the fraction of votes the other candidate will receive will be We shall take as a random variable with a uniform prior distribution over so that its mean is and variance is A few weeks before the election, two independent public opinion polls were conducted by two different pollsters. The first poll revealed that the candidate is likely to get fraction of votes. Since some error is always present due to finite sampling and the particular polling methodology adopted, the first pollster declares their estimate to have