Step-by-step explanation:
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Answer:
a)
So then we conclude that is an unbiased estimator of
So then we conclude that is an unbiased estimator of
b)
Step-by-step explanation:
For this case we know that we have two random variables:
both with mean
and variance
And we define the following estimators:
Part a
In order to see if both estimators are unbiased we need to proof if the expected value of the estimators are equal to the real value of the parameter:
So let's find the expected values for each estimator:
Using properties of expected value we have this:
So then we conclude that is an unbiased estimator of
For the second estimator we have:
Using properties of expected value we have this:
So then we conclude that is an unbiased estimator of
Part b
For the variance we need to remember this property: If a is a constant and X a random variable then:
For the first estimator we have:
Since both random variables are independent we know that so then we have:
For the second estimator we have:
Since both random variables are independent we know that so then we have: