The question that I'm am trying to answer is how to determine the expectation value of the standard deviation of 'k' closest samples to value 'A'. The standard deviation and mean of the underlying population is known, additionally, the distance metric is euclidean. I have been stuck for some time trying to figure out how to include the "nearest neighbors" aspect into the expectation calculation. It would also be helpful if anyone has seen similar problems, such as the expectation of standard deviation or calculating the stdev/mean including the 'nearest neighbors' aspect..
Blue line shows the distribution of the population. Each vertical line represents a sample from the population. The red line 'A' represents the sample, which the 'k' nearest samples (green lines) are compared to. The standard deviation and mean of the population is known. The value 'A' is not constant, but is always known (follows the same pdf as population distribution)