User:Schriste/Sensitivity (detectors)

A measurement or observation is a mixture of a signal,s, on top of a background, b. We shall assume that the background has been measured previously and that the error on that measurement is, sigma_b.

$$ obs = s + b $$ $$ s = obs - b $$ For the measurement of the signal to be statistically significant, we want signal to be K times larger than its error where K is the statistical significance. $$ s / \sigma_s = \frac{obs - b}{\sigma_{obs - b}} $$

error in the signal to be K times

The sensitivity of a detector system is a measure of the minimum detectable (i.e. statistically significant) signal for a telescope. It is typically measured in units of photons flux. It can be determined by solving the following equation $$ s - K \sqrt{\sigma_s^2 + 2 * \sigma_b^2} = 0 $$ where $$K$$ is the statistical significance of the signal, $$\sigma_s$$ is the statistical error in the signal, and $$\sigma_b$$ is the statistical error in the background. If the error on the signal follows Poisson statistics then $$ s - K \sqrt{s + \sigma_b^2} = 0 $$ which is equivalent to $$ s^2 - K^2 s - K^2 \sigma_b^2 = 0 $$ Using the quadratic equation the solution is $$ s = \frac{K^2 + \sqrt{K^4 + 4 K^2 \sigma_b^2}}{2} $$ If the background is well measured then the error in the background is

The sensitivity of a detector system is a measure of the minimum detectable (i.e. statistically significant) signal for a telescope. It is typically measured in units of photons flux. It can be determined by solving the following equation $$ s - K \sqrt{\sigma_s^2 + \sigma_b^2} = 0 $$ where $$K$$ is the statistical significance of the signal, $$\sigma_s$$ is the statistical error in the signal, and $$\sigma_b$$ is the statistical error in the background. If the error on the signal follows Poisson statistics then $$ s - K \sqrt{s + \sigma_b^2} = 0 $$ which is equivalent to $$ s^2 - K^2 s - K^2 \sigma_b^2 = 0 $$ Using the quadratic equation the solution is $$ s = \frac{K^2 + \sqrt{K^4 + 4 K^2 \sigma_b^2}}{2} $$ If the background is well measured then the error in the background is