Let X1, . . . , Xn denote a random sample from a N (µ, σ2) distribution,where the mean µ and the variance σ2 are both unknown so that the param-eter vector is given byθ = (µ, σ2)T .(i) Show that the joint p.d.f. of X1, . . . , Xn belongs to the two-parameterregular exponential family, stating what the canonical parameter isequal to in terms of θ

Let X1, . . . , Xn denote a random sample from a N(μ, σ2) distribution, where the mean μ and the variance σ2 are both unknown so that the param- eter vector is given by θ = (μ, σ2)T (i) Show that the joint p.d.f. of X1, . . . , Xn belongs to the two-parameter regular exponential family, stating what the canonical parameter is equal to in terms of θ. (ii) Use the result in (i) to derive the maximum likelihood (ML) estimate of θ, θˆ = (μˆ, σˆ2)T ,

how that the joint p.d.f. of X1, . . . , Xn belongs to the two-parameterregular exponential family, stating what the canonical parameter isequal to in terms of θ.

1. Suppose X = (X1, . . . , Xn) is a random sample from the Gamma distribution withshape α and rate θ, i.e.,fX (x) = θαΓ(α)xα−1e−xθ, x > 0, α > 0, θ > 0.Each Xi has expectation E(X) = α/θ, variance Var(X) = α/θ2, and momentgenerating function MX (t) = (1 − t/θ)−α.(a) Assuming both α and θ to be unknown, write down the log likelihood functionℓ(θ, α; X) and the corresponding score functions ∂ℓ∂θ and ∂ℓ∂α .(b) Verify that each score function has zero expectation.(c) Assuming α to be known:(i) Calculate the Cramer Rao Lower Bound (CRLB) for the variance of anunbiased estimator of θ.(ii) Is there any unbiased estimator of θ whose variance attain the CRLB?(iii) Show thatS = α − 1nnXi=1 1Xiis an unbiased estimator for θ. What is the MVU estimator for θ?(iv) Identify a change of parameter η = η(θ) for which there exists an unbiasedestimator with variance attained the CRLB.(v) For the parameter in the previous part, identify the MVU estimator whosevariance attains the CRLB. Compute this variance

The Poisson distribution truncated at zero has probability function given by f (x; θ) = exp(−θ)θx A(θ)x! , x = 1, 2, ... (θ > 0), where A(θ) = 1 − exp(−θ). Let X1, . . . , Xn be a random sample of size n from this truncated Poisson distribution. (i) Show that this distribution belongs to the regular exponential distri- bution and give its canonical parameter c(θ) and the corresponding sufficient statistic T (X1, ..., Xn)

Question 3. [9 Marks]. Recall that the exponential PDF has the form f (t; λ) = λe−λtwhere t, λ > 0 and λ is known as the rate parameter. Consider a job shop that consists of3 identical machines and 2 technicians. Suppose that, the amount of time each machineoperates before breaking down is exponentially distributed with rate parameter 0.1, and theamount of time a technician takes to fix a machine is exponentially distributed with rateparameter 0.4. Suppose that all the times to breakdown and times to repair are independentrandom variables and let X(t) be the number of machines which are operating at time t.(a) Determine the Q-matrix for this Markov process.(b) Write the forward equations involving the P ′0j (t), j = 0, 1, 2, 3, in terms ofpj (t) := P0j (t) = P(X(t) = j|X(0) = 0).(c) Obtain the equilibrium probabilities pj = limt→∞ pj (t).(d) What is the average number of busy technicians in the long-run? [Note: do not expectyour answer to be an integer.]