Description

Q1. Consider a Geiger counter in a nuclear power plant that measure the number of radiation counts. We observe 𝑛 readings from the counter and denote them as 𝑥1, 𝑥2, ⋯ , 𝑥𝑛. It is known that the number of radiation counts follows a Poisson distribution with parameter 𝜆.
𝜆𝑥𝑒−𝜆
𝑃(𝑥; 𝜆) = , 𝑥 = 0,1,2, ⋯
𝑥!
Find the MLE estimate of 𝜆 based on the observed values 𝑥1, 𝑥2, ⋯ , 𝑥𝑛.

Q2. Consider the following 20 data samples generated from a Poisson distribution:
𝑥1 𝑥2 𝑥3 𝑥4 𝑥5 𝑥6 𝑥7 𝑥8 𝑥9 𝑥10 𝑥11 𝑥12 𝑥13 𝑥14 𝑥15 𝑥16 𝑥17 𝑥18 𝑥19 𝑥20
3 1 2 1 2 1 0 2 1 0 5 6 3 2 4 4 0 5 5 3

Plot the MLE estimate for the parameter 𝜆 as a function of the number of samples (i.e., plot the MLE estimate for 𝜆 when you consider only 𝑥1, only 𝑥1 and 𝑥2, only 𝑥1, 𝑥2 and 𝑥3, and so on till you consider all 20 data points).