
I'm a late learner of science & maths.
These problems origin from Asian examinations.
Do you know any books discussing these kind of problems?
Sample queations
Q1
The mean and variance of a finite population P = where
, are defined as follows:
Samples of size 2, , where , are drawn at random from P. the mean of and is denoted by . These means then form a set
a) Show that
b) Show that the mean of all the elemeants of S is equal to
c) Show that the variance of all the elements of S is equal to .
hence deduce that if , this variance tends to .
solution
a)
b)
c)
variance =
as
we have
variance =
as , variance

here is another one without solution
Q2
let be observations of variable X and be observations of a variable . Let , , Var(X) = , Var(Y) =
a) prove that , where a, b are constants.
b) prove that Var(aX+b) = Var(X), where a, b are constants.
c) prove that Var +Var = Var(X) + Var(Y)
d) given the following observations on (X, Y), verify the equality in (c)
e)Let Z= aX +bY and W = cX +dY so that in matrix notation
Show that Var(Z) +Var(W) = Var(X) +Var(Y)
if

Q3
The mean and the variance of a finite production A = are defined as follows:
and
Samples of size 2, , are randomly drawn from A with replacement so that i=j is allowed. he mean of the sample is defined as These sample means then form a set S =
a) Show that
b) Show hat
i) the mean of all the elements of S is
ii) the variance of all the elements of S is
c) Suppose A =
Find the mean and variance of all the elements of S.
d) Suppose A =
Using (c), find the mean and variance of all the elements of S.
solution
a)
b)
i) mean of all the elements of S
(S has elements)
ii) Variance of all the elements of S
= Var(\bar{x_{ij}}= (S has elements)
b) ii) alternative method
c) mean =5.5
variance = 8.25
, Var( ) =
d) Using (c), mean =
Variance =
. Var =