Suppose that the number of expensive goods X sold in a shop over 24 days, is Poisson random variable with rate 240, i.e. X Poisson (240), where > 0 is the expected number of sales per day and is the unknown parameter that we would like to estimate. Suppose further that can take three possible values 0₁ = 1/2, 0, 1/4 and 0₁ = 1/8, with prior probabilities 0.2, 0.5 and 0.3, respectively. Suppose now that we observe that x=10 expensive goods were sold in the last 24 days. (a) Write down the likelihood function for and find the MLE of 0. (b) Given the observed data 2 = 10, what is the posterior distribution of 0, p(0 | x= 10)? (c) What is the posterior mean for ? (d) What is the posterior standard deviation for 0? [Hint: You may use the fact if X is a random variable, then var(X) = E(X²) – [E(X)]²].

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roctor es ces formula and a similar problem. During the 2017-2018 flu season, a random sample of 100 hospitalizations related to flu was studied. It was later determined that 10 patients have died. A similar study during the 2018-2019 flu season revealed that in a random sample of 80 hospitalized cases related to flu, there were 9 deaths observed. At 10% significance level, test the claim that the percentage of deaths in 2017-2018 is different from the percentage of deaths in the 2018-2019 season. Procedure: Two proportions Z Hypothesis Test く Assumptions: (select everything that applies) Population standard deviations are known Population standard deviation are unknown The number of positive and negative responses are both 5 or more for each sample Sample sizes are both greater than 30 Independent samples Simple random samples Normal populations Population standard deviation are unknown but assumed equal Paired samples Step 1. Hypotheses Set-Up: HPP Hap-P 0 0 =0 where p and p: are…