Answer:
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
![E[Y] = E[\frac{1}{n}\sum_{i=1}^{n} Y_i] = \frac{1}{n}\sum_{i=1}^{n} E[Y_i] = \frac{n (\theta+0.5)}{n} = \theta + 0.5](https://tex.z-dn.net/?f=%20E%5BY%5D%20%3D%20E%5B%5Cfrac%7B1%7D%7Bn%7D%5Csum_%7Bi%3D1%7D%5E%7Bn%7D%20Y_i%5D%20%3D%20%5Cfrac%7B1%7D%7Bn%7D%5Csum_%7Bi%3D1%7D%5E%7Bn%7D%20E%5BY_i%5D%20%3D%20%5Cfrac%7Bn%20%28%5Ctheta%2B0.5%29%7D%7Bn%7D%20%3D%20%5Ctheta%20%2B%200.5)
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.