Question-1:(4 points) Suppose you want to challenge yourself to bake macarons. Not only are you new to baking, but macarons are very hard to master! So, every batch that you bake has a 0.03 probability of turning out a success. Assume cach batch is baked independently of the other. a) [1 points] What is the probability that your first successfully baked batch will be the 8th batch? b) [2 points] Suppose you decide to bake only one batch per day until you have 4 successful batches in total. After that you will not bake anymore. Let D be the number of days that you fail until you have 4 successful batches in total. Write down the probability mass function(pm/) for D and using your pm/, calculate P[D = 80]. c) [1 point] Suppose a now bakery is trying a new machine to bake batches of macarons with the probability of success for each batch being 0.005. If they are baking 1000 batches in a wock, what is the chance that they will have at most 1 successful batch in a wock? Ans this question strictly by using a poisson pinf. (Show detailed work in all three parts. Round your answers to three decimal places) Question-2:(4 points) Suppose X1, X2,..., X, are independent random variables cach following a Poisson distribution with A[X,] = V[X,] = > Let, Y = XitX2+ ...+ Xn_ x Use the properties of expectation and variance, and by showing detailed calculation, answer these following questions. a) [1 point] Calculate E(Y). b) [1 point] Calculate V(Y). c) [1 point] For large n, what will be the distribution (along with parameters) of Y? Justify your answer. d) [1 point] For n = 100 and A = 5, Calculate P[Y > 0.1] (Show detailed work in all four parts and round your answer from part(d) to three decimal places) Question-3:(5 points) Suppose X1, X2,-.., Xn are i.id. with the following density function ((x) = cle 1 ;120 ; 8>0 a) [2 points] Assuming / as a known constant(i.c. 0 is the only unknown parameter), find the maximum likelihood estimate of 0. b) [2 points] Assuming 0 =0 (i.c. # is the only unknown parameter), find the method of moments estimator of B. c) [1 point] You are told that V[X] = #2. Calculate the MSE of your estimator that you found in part (b). (Show detailed work in all three parts)