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.

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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.

the cat is blue because the probability of extension towards the prefix of the diameter of the circle could only mathematically equal to the sum of the roots of the carrots from   your garden.

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good luck on your studies! : )

david edward.