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].

A machine is built to make mass-produced items. Each item made by the machine has a probability of being defective. Given the value of 0, the items are independent of each other, where is unknown and would like to estimate. Suppose has for prior distribution a Beta(a, ß) distribution, where a > 0 and 3>0. The machine is tested by producing items until the first defective occurs. Suppose that the first 12 items are not defective but the y = 13th item is defective. (a) Write down the likelihood function for 0 and find the MLE of 0. (b) Given the observed data y = 13, what is the posterior distribution of 0, p(0 | y = 13)? Take a = 1 and ẞ= 19. (c) What are the parameters of the posterior distribution? (d) What is the posterior mean for 0? (e) What is the posterior standard deviation? =

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].