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Suppose I have a normal distribution, $N[\mu,\sigma]$, and I have a sample of size $n$. It is well know that the error (std deviation) of the mean is $\sigma/\sqrt{n}$.

Now suppose that my distribution is a mixture of signal (normal distributed) and a flat background $$\text{pdf}[x|\mu, \sigma, s, b] = \frac{sN[x|\mu,\sigma] + b U}{s+b} $$

suppose that the uniform distribution is much wider than the normal but in the signal window the background is important. Question: what is the error of $\mu$ now?

($s$ and $b$ are random variables, they can be assumed to be Poisson distributed)

Glen_b
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Ruggero Turra
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  • The solution given at http://stats.stackexchange.com/questions/16608/what-is-the-variance-of-the-weighted-mixture-of-two-gaussians applies equally well here, too: you only need to know that the mean and variance of $U$, which are easy to determine (or even look up). – whuber Oct 24 '13 at 21:38
  • @whuber: no, I think it is different. I want the error on $\mu$, not the error on the mean of the mixture. – Ruggero Turra Oct 24 '13 at 21:45
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    The SE of the mean is determined by the variance of the sample average, which is directly related to the variance of the underlying distribution. To understand this better you might want to research our site for threads concerning standard errors. – whuber Oct 24 '13 at 21:54
  • Your question used the same symbol for two different things. I edited to make the sample size $n$, so that you don't reuse the symbol you used to denote the distribution ($N$). – Glen_b Oct 25 '13 at 00:29
  • Making $s$ and $b$ random variables is a complication, in particular because the pdf now has a positive chance of being undefined (when both $s$ and $b$ are zero)! This new formulation of your question needs more careful thought. – whuber Oct 25 '13 at 02:05

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