a)
First we find the mean and variance of
:
Now, consider the following identity:
From the above equation we note that:
, thus
is not an unbiased estimator of
. [ANSWER]
b)
The formula for the sample variance is given by:
Thus, the expected value of the sample variance is given by:
Now, for
to be an unbiased estimator of
, it must satisfy:
Thus, k = 1/n is the value of k for which the estimator
is unbiased for
. [ANSWER]
c)
We know that:
Thus, both the estimators X1 and
are unbiased. Now, the variances of the two estimators are given
by:
From above we note that:
Since,
has lesser variance than X1,
is more efficient than X1. [ANSWER]
For any queries, feel free to comment and ask.
If the solution was helpful to you, don't forget to upvote it by clicking on the 'thumbs up' button.
3. Using a long rod that has length y, you are going to lay out a...
Using a long rod that has length , you are going to lay out a
square plot in which the length of each 2. side is . Thus the area
of the plot will be However, you do not know the value of , so you
decide to make n independent measurements X1;X2; :::;Xn of the
length. Assume that each Xi has mean 2. (unbiased measurements) and
variance o^2.
Question C only, Now if I estimate u using both X1...
3. Using a long rod that has length y, you are going to lay out a square plot in which the length of each side is p. Thus the area of the plot will be ?. However, you do not know the value of p, so you decide to make n independent measurements X1, X2, ..., Xn of the length. Assume that each X, has mean y (unbiased measurements) and variance o?. a) Is X2 unbiased for ? why or...
3. Using a long rod that has length y, you are going to lay out a square plot in which the length of each side is p. Thus the area of the plot will be j?. However, you do not know the value of u, so you decide to make n independent measurements X1, X2, ..., Xn of the length. Assume that each X; has mean u (unbiased measurements) and variance o?. a) Is Xunbiased for u? ? why or...
3. Using a long rod that has length p, you are going to lay out a square plot in which the length of each side is u. Thus the area of the plot will be u?. However, you do not know the value of p, so you decide to make n independent measurements X1, X2, ..., Xn of the length. Assume that each X; has mean u (unbiased measurements) and variance o2. a) Is X2 unbiased for u? ? why...
Using a long rod that has length μ you are going to lay out a square plot in which the length of each side is Thus the area of the plot will be However, you do not know the value of μ, so you decide to make n independent measurements X1,X2, ,X, of the length. Assume that each Xi has mean μ and variance σ Show that X 2 is not an unbiased estimator for μ2 What is the bias?...
can anyone help me with #3, especially c) thank you
3. Using a long rod that has length, you are going to lay out a square plot in which the length of each side is . Thus the area of the plot will be. However, you do not know the value of , so you decide to make n independent measurements X1, X2, ..., X, of the length. Assume that each Xhas mean (unbiased measurements) and variance o?. a) Is...
Now if I estimate y using both Xi and X, which estimator is better in terms of efficiency? (3] 3. Using a long rod that has length y, you are going to lay out a square plot in which the length of each side is p. Thus the area of the plot will be u2. However, you do not know the value of , so you decide to make n independent measurements X1, X2, ..., Xn of the length. Assume...
Using a long rod that has length , you are going to lay out a
square plot in which the length of each 2. side is . Thus the area
of the plot will be However, you do not know the value of , so you
decide to make n independent measurements X1;X2; :::;Xn of the
length. Assume that each Xi has mean 2. (unbiased measurements) and
variance
1. A large insurance agency services a number of customers who have...
Using a bar of length μ, you are going to lay out a square plot in which the length of each side will be length μ. Thus the area of the plot will be μ2. However, you do not know the value of μ and so you decide to make 15 independent measurements of the bar, {Xi : i = 1, … , n}. Assume that each Xi has mean μ (unbiased measurement) and variance σ2 = 0.4.(a)Find the bias of the random variable X_2 as an estimator for μ2.(b)For what value of k is the estimator X_2 − kS2 unbiased...
3. (5 marks) Let U be a random variable which has the continuous uniform distribution on the interval I-1, 1]. Recall that this means the density function fu satisfies for(z-a: a.crwise. 1 u(z), -1ss1, a) Find thc cxpccted valuc and the variancc of U. We now consider estimators for the expected value of U which use a sample of size 2 Let Xi and X2 be independent random variables with the same distribution as U. Let X = (X1 +...