I have two normally distributed random variables that add together to form a third normally distributed variable. The means in all cases are 0. Thus, the standard deviations of the first 2 variables add in quadrature:
I want to measure directly
sigma_a, but the instrument used adds
sigma_b noise. Currently I measure (separately)
sigma_c, then solve for
The problem is when the instrument noise floor dominates the signal of interest. In the extreme case when this occurs,
sigma_c (the signal of interest plus noise) approximately equals
sigma_b (the noise), and my estimate of
sigma_a (the signal of interest) returns 0, which is incorrect (e.g. I've reached the noise floor of the instrument).
Is it possible to somehow quantify the error in
Ultimately I need to determine whether
sigma_a is below a compliance limit
L with some amount of certainty. Is this possible given the above data?
Is the following true?
"If n samples were taken to compute each of
sigma_b, then the standard error for the signal of interest is
sigma_a/sqrt(n). Thus, one can be 95% confident that the sample mean of distribution
a is within plus or minus 1.96*(standard error) from the true population mean."