Question

a) (8 points) The number of breakdowns Y per day for a certain machine is a Poisson random variable with mean 1. The daily cost of repairing these breakdowns is given by C = 3Y2. If Y1, ..., Yn denote the observed number of breakdowns for n independently selected days, find and MVUE for E(C). b) (7 points) A random sample Y1, ..., Yn is selected from a population of Uniform(0,0) random variables. Is the estimator Ô = 27 a consistent estimator for 0. Justify clearly why or why not.

Answer 1

a) Given: y~ Poission (A) Y is per day C = 3y2 By the formula elax]e aflx) where no. of breakdowns for a machine. YuY₂... Yn be the observed number of breakdowns for n in dependently selected days. C is daily costs of repaining breakdowns.ie, 3y2 then E (c)= E(37²) = 3ELY) As we know that y apoission (1) > ELY) = Var (Y) = x Again, var(y)= Ely?)- Ely)? 2) ElyY= Ely) + Var (y) =) Ely)= x++ Hence, E(C)= 3 Elyy = 3(x²+x)= 3x²431 . To Find: MVUE of E(c)= 3 1²+ 37. Solution: To proceed ceed further, we should know some concept exponential like 'One Parameter, fainly distribution (opef), Lehmany-Schefte them,

Note: As we know that The necesso essary and sufficient condition for consistency is E(To) ii) VarlTn) where Tn is the statistic. i) as n to as no as Hence, E (27) var (27) is the consistent estimator of o. ay nto = ) 27