Estimating the Number of Volumes in a Library Using Two Proposed Estimators

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In summary, the library has three books but it is not known how many volumes there are altogether in the set. Two estimators of n are proposed: Y=2(3X3-X1) and Z=2X2-1. Y must be 4 and Z must be 3.
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Homework Statement



A library has been given 3 books. these books carry volume numbers X1, X2 and X3 where
X1<X2<X3. But it is not known how many volumes there are altogether in the set. Suppose there are n volumes, numbered 1,2,3,...n in the set, and the 3 volumes in the library are regarded as a random sample from this total of n. Two estimators of n are proposed:

Y=2(3X3-X1)
Z=2X2-1
Consider the case where n is 3. show that the value of Y must be 4 and that the value of Z must be 3.

Homework Equations





The Attempt at a Solution



I know i have to find X1 and X2 and X3 but i don't know how to start.

Thank you!
 
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  • #2


There is only one possible set of volume numbers for the case n=3. What is it?
 
  • #3


Yes, that is 1,2 and 3

but how does help to find X1, X2 and X3
(sorry if this is a stupid question)
Thank you.
 
  • #4


You are picking an ordered set of three elements from an ordered set of n elements. For example, with n=4 there are three possible selections: (X1,X2,X3) = (1,2,3), (1,2,4), or (2,3,4). With n=5, the number of combinations jumps to 6, 10 for n=6.

There is only one possibility with n=3, so there is only one possibility for the estimators.
 
  • #5


Reread the problem and THINK. The library has three volumes of a set. The set only HAS 3 volumes! What are they numbered?

However, Y= 2(3X3-X1) is NOT 4. Did you mean Y= (3X3-X1)/2?
 
  • #6


D H said:
You are picking an ordered set of three elements from an ordered set of n elements. For example, with n=4 there are three possible selections: (X1,X2,X3) = (1,2,3), (1,2,4), or (2,3,4). With n=5, the number of combinations jumps to 6, 10 for n=6.

There is only one possibility with n=3, so there is only one possibility for the estimators.

Why isn't (1,3,4) a possible selection??
 
  • #7


Have you ever seen a 3 book set numbered "volume 1", "volume 3", "volume 4"? Don't you think that would cause people to wonder where "volume 2" was?
 
  • #8


Yeah but how come there could be such thing as (1,2,4) ...where's vol 3?? :)

I'm sorry i don't think I am getting the point of how to get X1, X2 and X3.
 
  • #9


There is no volume 4. Reread the problem statement that you supplied. Think.
 
  • #10


Let's do this one volume at a time.

X1 is the smallest volume number (since you are told X1<X2<X3). What must it be?

PS: Remember we are dealing with the case whee n = 3, i.e., the series has only 3 volumes.
 
Last edited:
  • #11


sara_87 said:
Yeah but how come there could be such thing as (1,2,4) ...where's vol 3?? :)

I'm sorry i don't think I am getting the point of how to get X1, X2 and X3.
Who said there could be "such a thing as (1, 2, 4)"? If there are only 3 volumes in the set, then they are numbered volume 1, volume 2, volume 3!
 
  • #12


Oh ok ok ok, so there's only 3 volumes X1 X2 and X3 and n=3 so this means X1=1
X2=2 and X3=3

is that right?
 
  • #13


Yes.
 

1. What is an estimator in mathematics?

An estimator in mathematics is a statistical method or formula used to estimate an unknown quantity or parameter based on a sample of data. It is commonly used in statistics and probability theory to make predictions or inferences about a population based on a limited amount of observed data.

2. How is an estimator different from an estimate?

An estimator is a mathematical concept or formula, while an estimate is the actual value calculated using that estimator. An estimator is used to make an estimate, which is an approximation or prediction of the true value of a parameter.

3. What are the properties of a good estimator?

A good estimator should be unbiased, meaning that on average it should produce estimates that are close to the true value of the parameter. It should also be consistent, meaning that as the sample size increases, the estimate should approach the true value. Additionally, a good estimator should have low variability and be efficient, meaning that it should have a small variance compared to other estimators.

4. How do you choose the best estimator for a given problem?

The choice of estimator depends on the specific problem and the type of data being analyzed. It is important to consider the properties of the estimator, such as bias, consistency, and efficiency, as well as the assumptions and limitations of the data. In some cases, multiple estimators may be used and compared to determine the best option.

5. How do you assess the accuracy of an estimator?

The accuracy of an estimator can be assessed by comparing the estimated value to the true value of the parameter. This can be done using measures such as mean squared error or confidence intervals. It is also important to consider the variability and potential bias of the estimator when assessing its accuracy.

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