## Question

###### Let Xi,Xz, variables isXn "4 Exponential(8). The probability density function for these randomf(r18)e-5 , 1 > 0; 8 > 0.Find the maximum likelihood estimator for 8.b. Compare the MLE to the estimator 8 (Xi + Xn)/2. Which do you recommend? Explain. You can use the following facts: E(X;) = 8 and Var(X;) = 82 , for anyConstruct a 100(1 _ a)% C.I. for B using the pivotal quantity (2nBxLe) /8. You can use the fact that (2nBuLe) /8 xn=Is 1/8 MLE unbiased for 1/82 Prove youT claim. You ca

Let Xi,Xz, variables is Xn "4 Exponential(8). The probability density function for these random f(r18) e-5 , 1 > 0; 8 > 0. Find the maximum likelihood estimator for 8. b. Compare the MLE to the estimator 8 (Xi + Xn)/2. Which do you recommend? Explain. You can use the following facts: E(X;) = 8 and Var(X;) = 82 , for any Construct a 100(1 _ a)% C.I. for B using the pivotal quantity (2nBxLe) /8. You can use the fact that (2nBuLe) /8 xn= Is 1/8 MLE unbiased for 1/82 Prove youT claim. You can use the facts Xi-IXX; Gamma(n, 8) and E [(Ei-1X;)*] (8*T(n + k)) /T(n) when ~n and Ci-IX; Gamma(n, B).

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