Question 387694
For now I'll just do 2 and 3(?).
A large bag contains counters. Sixty per cent of the counters have the number 0 on them and
40% have the number 1.
(a) Find the mean μ and the variance σ2 for this population of counters.
A random sample of size 3 is taken from this population.
(b) List all possible samples.
(c) Find the sampling distribution for the mean X =
3
1 2 3 X  X  X
where X1, X2 and X3 are
the three variables representing samples 1, 2 and 3.
(d) Hence find E ( X ) and Var( X ).Find the sampling distribution for the mode M.
(f) Hence find E (M) and Var(M). 
---------------------------------------------------------------------------------
(a) It is binomially distributed, with the following distribution:
X       | 0        1
-------------------
p(X )  | 0.60    0.40

Then {{{mu(X )= E(X) = 0*0.60 + 1*0.40 = 0.40}}}
{{{(sigma(X))^2 = E(X^2) - (E(X))^2 = 0^2*0.0 + 162*0.40 - 0.40^2 = 0.40*0.60 = 0.24}}}
(b)Sample space = {000,001,010,011,100,101,110,111}.
(c) sampling distribution of the mean X`
X`    --->      0         1/3          2/3          1
p(X`) --->     27/125     54/125       36/125       8/125

(d) Then {{{mu}}}(X`) = {{{18/125 + 24/125 + 8/125 = 2/5 = 0.4}}}, and 
{{{sigma^2}}}(X`) = {{{E(X^2) - (E(X))^2  = 6/25 - 4/25 = 2/25}}}.
(e) Sampling distribution of the mode:
M    --->     0             1
p(M) --->     81/125         44/125
The mode is 0 if there are 3 or 2 0's, and 1 is the mode if there are 3 or 2 1's.
(f)  {{{mu(M) = E(M)  = 44/125}}}, while {{{sigma^2(M) = E(M^2)- (E(M))^2 = 44/125 - (44/125)^2 = 3564/15625}}}.