Reading 4: Maximum Likelihood Estimation#
For the class on Wednesday, January 22nd
Reading assignments#
Questions#
Submit your answer on Canvas. Due at noon, Wednesday, Jan 22nd.
List any concepts that confuse you from your reading. Explain why they confuse you. If nothing confuses you, briefly summarize what you have learned from this reading assignment.
You have \(N\) independent measurements \(k_1, k_2, \dots, k_N\) that you believe come from one particular Poisson distribution with an unknown mean \(\lambda\).
(a) What is the log-likelihood function \(\ln \mathcal{L}(\lambda)\) given this data set \((k_1, k_2, \dots, k_N)\) ?
(b) Derive the Maximum Likelihood Estimation for \(\lambda\), denoted as \(\hat{\lambda}\). Show your steps.
(c) The observed Fisher Information for \(\hat{\lambda}\) is defined as \( - \frac{d^2}{d\lambda^2} \ln \mathcal{L}(\lambda) \big|_{\lambda = \hat{\lambda}}\). Calculate the observed Fisher Information in this case. Show your steps.
Hint
The PMF of a Poisson distribution with a mean of \(\lambda\) is \(P(k) = \frac{\lambda^k e^{-\lambda}}{k!}\)
You flipped a coin \(N\) times, and you got heads \(K\) out of \(N\) times. What is the likelihood that this coin is a fair coin (express in terms of \(N\) and \(K\))? Show your steps.