We have two datasets $($CW$2\_$spectrum$1.$dat and CW$2\_$spectrum$2.$dat$)$ containing spectra with $10,000$ bins, from $100$

to $200$

microns. You can use np$.$loadtxt to load these files. Maybe it's from a far$-$infrared spectrograph, or maybe I just wanted to make the numbers simple. The first line is easy to see, but the second is not.

Each spectrum contains a single spectral line, but also featureless Gaussian noise with an unknown amplitude. The line has a shape refered to as Cauchy or Lorentzian. There are three parameters $-$ the central wavelength $0$

$,$ the width $$

$,$ and the amplitude $$

$.$

$()=1+(0)2$

You can assume that the width is less than $10$ micron, and the width of the second line is narrower than the first. The wavelength could be anywhere but is not too close to the edge.

a$)$ Plot the data.

b$)$ Set up an MCMC model for the line plus the noise. Use it to infer the parameters for the two datasets.

c$)$ Check that your models converged $($i$.$e$.$ the inference worked well$).$ Write a sentence or two about how you know they converged, and include a plot.

d$)$ Plot the posteriors of the parameters. Include the amount of noise if you can.

e$)$ Summarize your results. How would you briefly describe what you know about the wavelength and width of the line, including uncertainty?