Estimate μ, Variance Unknown

Abstract

Consider X a random variable with probability density function N(μ, σ 2), which is the normal probability density with unknown mean μ and unknown variance σ 2. To estimate μ we obtain a random sample X 1, ..., X n from N(μ, σ 2). Suppose the mean of this random sample turns out to be $$\mathop s\nolimits^2 = {\sum\limits_{i = 1}^n {\left( {\mathop x\nolimits_i - \overline x } \right)} ^2}/\left( {n - 1} \right) $$ (4.2) , which is a crisp number, not a fuzzy number. Also, let s 2 be the sample variance. Our point estimator of μ is \(\overline{x}\). If the values of the random sample are x1, ..., x n then the expression we will use for s 2 in this book is $${{s}^{2}}={{\sum\limits_{i=1}^{n}{({{x}_{i}}-\overline{x})}}^{2}}/(n-1).$$ (4.1) We will use this form of s 2, with denominator (n−1), so that it is an unbiased estimator of σ 2.