How Is the Variance Used as Weight in Half-Life Error Analysis?

[schniefen]

Homework Statement

Conceptual question on error analysis (see attached image).

Relevant Equations

No equations.

What is meant by the constant background and how would one deduce the half-life IF not from the fit?

 

[Mark44]

Homework Statement:: Conceptual question on error analysis (see attached image).
Relevant Equations:: No equations.

What is meant by the constant background and how would one deduce the half-life IF not from the fit?

View attachment 274317

Does the textbook or other resource talk about the term "constant background"? Does it give any examples of what they're talking about?

 

[schniefen]

It has the following section:

How is the statistical error related to the standard error? Are they the same? I have not read a lot of error analysis, although this is probably a very basic question. I'm familiar with estimators and the like, and probability theory in general. "Constant background" is not mentioned.

 

[Mark44]

From wikipedia ( https://en.wikipedia.org/wiki/Standard_error )

The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error of the mean (SEM).

The standard error of the mean is $\sigma_x = \frac {\sigma}{\sqrt n}$
What's the underlying distribution here? In a normal distribution the parameters are the mean ($\mu$) and standard deviation (\sigma). In this kind of distribution, the parameters are related, as they seem to be in the text you quoted.

From that text, it appears that they are assuming a Poisson distribution, in which the mean ($\mu$) and variance ($\sigma^2$) are equal. So $\mu = \lambda = \sigma^2$, or $\sigma = \sqrt{\lambda}$. In the text, $\lambda = N$.

I don't understand their terminology of "the value from the fit as the weight."

 

[schniefen]

It must be the Poisson distribution as you say, since "number of counts in one channel" most likely refers to some kind of decay.

How does one use the variance $\sigma^2=N$ as the "weight in a fit"?