The reading on a voltage meter connected to a test circuit is uniformly distributed over the interval (θ, θ + 1), where θ is the true but unknown voltage of the circuit. Suppose that Y1,Y2,...,Yn . a Show that Y is a biased estimator of θ and compute the bias. b Find a function of Y that is an unbiased estimator of θ.

Step-by-step explanation:

The complete question is

The reading on a voltage meter connected to a test circuit is uniformly distributed over the interval (θ, θ + 1), where θ is the true but unknown voltage of the circuit. Suppose that Y1,Y2,...,Yn denotearandomsampleofsuchreadings.Let Y be the sample mean. a) Show that Y is a biased estimator of θ and compute the bias. b Find a function of Y that is an unbiased estimator of θ.

Recall that an unbiased estimator Y of a parameter is  a function of a random sample for which we have that

. When this is not the case, the quantity is called the biased of the estimator.

Recall that for each i, is uniformly distributed on the interval , then .

Then, using the linear property of the expeted value, we have that

So, Y is a biased estimator of [tex]\theta [/tex} and the bias is 0.5.

b) We can easily obtain an unbiased estimator of theta by simply substracting the bias to the biased estimator, that is Y-0.5 is an unbiased estimator of the parameter theta.

c. (3,2,6) is the answer.