Answer to Question #60652 in Statistics and Probability for esita

Question

9. a) If x ≥1 is the critical region for testing 2 : H0 θ = against the alternative θ = 1, on
the basis of the single observation from the population,
f ( x, θ) = θexp(−θ, x), 0 ≤ x < ∞
Obtain the values of type I and type II errors. (3)
b) For the geometric distribution,
( , ) 1( ) , ,1 ,2 , 0 1
1
θ = θ − θ = < θ < f x
x−
x K
Obtain an unbiased estimator of /1 θ. (3)
c) Given a random sample X X Xn
, , ,
1 2 K from a normal ) ( ,
2 N µ σ distribution,
examine unbiasedness and consistency of (i) X for µ , (ii) ∑ −
2
( )
1
X X
n
i

for 2 σ .

Answer 1

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