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Suppose that each of two statisticians A and B must estimate a certain
parameter theta whose value is unknown (theta > 0). Statistician A can observe
the value of a random variable X which has a gamma distribution with parameters
alpha and beta, where alpha = 3 and beta = theta; and statistician B can observe
the value of a random variable Y which has a Poisson distribution with mean
2(theta). Suppose that the value observed by statistician A is X = 2 and the
value observed by statistician B is Y = 3. Show that the likelihood functions
determined by these observed values are proportional, and find the common value
of the M.L.E. of theta obtained by each statistician. The M.L.E. is
closest to:
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