Editing Standard error (section)

===Independent and identically distributed random variables with random sample size===
There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size <math>N</math> is a random variable whose variation adds to the variation of <math>X</math> such that,<math display="block">\operatorname{Var}(T) = \operatorname{E}(N)\operatorname{Var}(X) + \operatorname{Var}(N)\big(\operatorname{E}(X)\big)^2</math><ref>{{ cite book | last1 = Cornell | first1 = J R | last2 = Benjamin | first2 = C A | title = Probability, Statistics, and Decisions for Civil Engineers | publisher = McGraw-Hill | location = NY | year = 1970 | isbn = 0486796094 | pages = 178–179 }}</ref>
which follows from the [[law of total variance]].

If <math>N</math> has a ''[[Poisson distribution]]'', then <math>\operatorname{E}(N)= \operatorname{Var}(N)</math> with estimator <math>n = N</math>. Hence the estimator of <math>\operatorname{Var}(T)</math> becomes <math>nS^2_X + n\bar{X}^2</math>, leading the following formula for standard error: 
<math display="block">\operatorname{Standard~Error}(\bar{X})= \sqrt{\frac{S^2_X + \bar{X}^2}{n}}</math> 
(since the standard deviation is the square root of the variance).