Question

An urn is filled with balls, each numbered n = 0, 1, or 2. The average of n is <n> = 2/7. Calculate the probabilities p₀, p₁, p₂, which yields the maximum uncertainty. find the expectation value, based these probabilities, of <n³> - 2<n>.

Answer 1

Solution:

Let us first calculate the expected value of n and is as follows

<n>=p₀*0+p₁*1+p₂*2

=p₁+2p₂

But the average of n is <n> = 2/7.

So, p₁+2p₂=2/7-----------------------------------1

This is one of the constraints for maximizing the uncertainty function.

We know p₀+p₁+p₂=1, this is the second constraints.---------------------------2

Using these we have, S^'=S/C-(p₁+2p₂-2/7)-(p₀+p₁+p_2-1)

By taking derivative with respect to and and then equating it to zero, we have

Log p₀+1- =0-----------------3

Log p₁+1- -  =0-------------------4

Log p₂+1-2 -  =0-------------------------5

By solving equation 1,2 3,4 anf 5, we have p₀=15/21 , p₁=4/21 and p₂=1/21

Now we can find <n³> - 2<n>=p₀*0+p₁*1³ +p₂*2³-2<n>

=p₁+8 p₂-2(2/7)

=(4/21)+8(1/21)-(4/7)=0