PHALIN THAKKAR / PORTFOLIO

PHALIN THAKKAR

Where the reference intensity belongs

What quantity is the denominator, and what uncertainty does it share with the numerator?

Note 2 · Explanatory note

Analytical result · Hypothetical examples are not physical measurements.

A normalized reading needs a defined source state and detector setup. Distinguish a separately measured aligned reference from a fitted amplitude or an assumed maximum.

Use the existing illustrative counts: reference R = 240, signal X = 125 and shared background B = 10.

r=X−BR−B=115230=0.50.r=\frac{X-B}{R-B}=\frac{115}{230}=0.50.

Now suppose the raw readings have independent standard uncertainties σX = σR = 5 counts and σB = 2 counts. These are stipulated uncertainties in a hypothetical example, not Poisson uncertainties inferred from those counts.

The derivatives are:

∂r∂X=1R−B,∂r∂R=−X−B(R−B)2,\frac{\partial r}{\partial X}=\frac1{R-B},\qquad \frac{\partial r}{\partial R}=-\frac{X-B}{(R-B)^2},
∂r∂B=X−R(R−B)2.\frac{\partial r}{\partial B}=\frac{X-R}{(R-B)^2}.

First order propagation gives:

σr2=∑z=X,R,B(∂r∂z)2σz2,\sigma_r^2=\sum_{z=X,R,B} \left(\frac{\partial r}{\partial z}\right)^2\sigma_z^2,

so σr ≈ 0.0247.

The same background estimate appears in both corrected readings. Treating those corrected quantities as independent would miss their covariance. Using the independent raw quantities makes that shared effect explicit.

Continue the reasoning

Finding the hidden direction

All twelve notes

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