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

[sara_87]

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!

 

[D H]

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

 

[sara_87]

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.

 

[D H]

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.

 

[HallsofIvy]

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?

 

[sara_87]

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??

 

[HallsofIvy]

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?

 

[sara_87]

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.

 

[D H]

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

 

[Gokul43201]

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.

 

[HallsofIvy]

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!

 

[sara_87]

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?

 

[Gokul43201]

Yes.