Unbiased Estimator for True Voltage: Find & Show Bias

[sara_87]

Homework Statement

The reading on a voltmeter connected to a test circuit is uniformly distributed on the interval (a, a+1) where a (unknown) is the true voltage. Let Y1, Y2, Y3,...,Yn denote a random sample of such readings.
show that Y(bar) is a biased estimator for a.
and find an unbiased estimator for a expressed in terms of Y(bar)

Homework Equations

The Attempt at a Solution

i have no idea how to do thus question so any help would be v much appreciated.
Thank you

 

[mighty2000]

for your post. I am happy to help.

First, let's define some terms and clarify the question. A voltmeter is a device used to measure voltage, and in this case, it is connected to a test circuit. The readings on the voltmeter are uniformly distributed on the interval (a, a+1), where a is the true voltage. This means that any reading on the voltmeter is equally likely to occur within this interval.

Now, let's consider the random sample of readings, Y1, Y2, Y3,...,Yn. These readings are a sample from the true voltage a, and we can calculate the average of these readings, denoted as Y(bar). This is known as the sample mean and is given by the formula Y(bar) = (Y1+Y2+Y3+...+Yn)/n.

The question asks us to show that Y(bar) is a biased estimator for a. A biased estimator is one that, on average, does not equal the true value it is estimating. In this case, Y(bar) is biased because it is not equal to the true voltage a. This is because the readings on the voltmeter are not evenly distributed around the true voltage a, but rather around the interval (a, a+1). Therefore, the sample mean Y(bar) will tend to overestimate the true voltage a.

To find an unbiased estimator for a, we can use a simple adjustment to Y(bar). If we subtract 1/2 from each reading in the sample, the new sample mean, denoted as (Y1-1/2)+(Y2-1/2)+...+(Yn-1/2)/n, will be an unbiased estimator for a. This is because now the readings are evenly distributed around the true voltage a, as the interval (a, a+1) is shifted to (a-1/2, a+1/2). Therefore, the sample mean will now give a more accurate estimate of the true voltage a.

In conclusion, Y(bar) is a biased estimator for a, and an unbiased estimator for a can be found by subtracting 1/2 from each reading in the sample and calculating the new sample mean. I hope this helps clarify the question and provides a solution to the problem. Let me know if you have any further questions. Good luck with your studies!