Question

Photon counts and source strength. an astronomer measures the photon flux from a distant star using a very sensitive instrument that counts single photons. after a given exposure time t, the instrument has collected rˆ photons. one can assume that the photon counts, rˆ, are distributed according to the poisson distribution. the astronomer wishes to determine λ, the emission rate of the source.

what is the likelihood function for the measurement?
(a) The astronomer strives to discern λλ through meticulous measurement of photon counts, employing the likelihood function in this astronomical pursuit.
(b) The astronomer seeks λλ by counting photons, utilizing the likelihood function to derive accurate emission rates.
(c) Utilizing the likelihood function, the astronomer calculates λλ, the emission rate, from the measured photon counts in a dedicated observation period.
(d) In the quest for λλ, the astronomer employs the likelihood function, relying on photon count measurements for accurate determination.

Answer 1

The likelihood function for the measurement is obtained by applying the Poisson distribution, which allows the astronomer to calculate the likelihood of observing a certain number of photons given a specific emission rate λ. By maximizing the likelihood function, the astronomer can estimate the most probable emission rate of the source.

Step-by-step explanation:

The likelihood function for the measurement is obtained by applying the Poisson distribution, which is a probability distribution that describes the number of events occurring in a fixed interval of time or space. In this case, the events are photon counts from the distant star. The likelihood function is given by:

L(λ|r) = (e^(-λ) * λ^r) / r!

Where λ is the emission rate of the source and r is the number of photons collected by the instrument after the exposure time. This function allows the astronomer to calculate the likelihood of observing a certain number of photons given a specific emission rate λ. By maximizing the likelihood function, the astronomer can estimate the most probable emission rate of the source.