A Pause for Reflection — Question 4

Question 4

Competing angular models can be compared with normalized probabilities

Two ideal models for x=cos(theta) on [-1,1] have densities $f_A=1/2$ and $f_B=3(1+x^2)/8$. Ignore detector acceptance.

1. Integrate each density over the central interval |x|<1/2.

2. For 1600 events, find predicted central counts and the difference in units of the binomial standard deviation under A.

Question 4 - Solution

Competing angular models can be compared with normalized probabilities

(a) Method and calculation

$$\boxed{P_A=\int_{-.5}^{.5}\frac12\,dx=.5}.$$$$\boxed{P_B=\frac38[x+x^3/3]_{-.5}^{.5}=13/32=.40625}.$$

(b) Interpretation and further calculation

$$\boxed{N_A=800,\quad N_B=650,\quad\Delta N=150}.$$$$\boxed{\sigma_A=\sqrt{1600(.5)(.5)}=20,\quad\Delta N/\sigma_A=7.5}.$$

Physical check

This is an expected separation for specified ideal models, not an observed discovery significance with systematic effects.