Let X_1, X_2, ..., X_n be a random sample from N(0, Θ). The variance Θ is unknown. Derive the most powerful test for H_0 : Θ = Θ_0 against H_1 : Θ = Θ_1, where Θ_1 is a fixed positive number. Is it a UMP test?

3Medium15 marks

Stating the underlying assumptions, find the expression of the probability of happening n events in the time interval (0, t) on Poisson process. Hence obtain the distribution of inter-arrival times.

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