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Suppose we need to divide people into two groups A and B, the first person will be assigned to either of the group with probability $0.5$, from the second person, the assignment will be done based on the following rule:
If the number of person in the group A is $a$ and the number of people in the group B is $b$, then for the $(a+b+1)_{th}$ assignment, the probability of assignment to Group A is $\frac{b}{a+b}$ and the probability of assignment to Group B is $\frac{a}{a+b}$.

After finishing the assignment of $n$ person, we define $X_{n}$ as the random variable for the number of person in Group A. Apparently we have $E(X_{n}) =\frac{n}{2} $, also I assume that the variance $V(X_{n})$ can be calculate in a recursive way.

My question is how to prove the following recursive formula:
$V(X_{n}) = \frac{n-3}{n-1} V(X_{n-1}) + \frac{1}{4}$

By applying conditional probability, I was able to calculate the expectation in a recursive way as below:
$E({X_{n+1}|X_{n}}) = P({X_{n+1} = X_{n} + 1|X_{n}})(X_{n}+1) + P({X_{n+1} = X_{n} |X_{n}})X_{n} = (1-\frac{1}{n})X_{n} + 1$

$E({X_{n+1}}) = E_{X_{n}}(E({X_{n+1}|X_{n}})) = (1-\frac{1}{n})E(X_{n}) + 1$

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  • $\begingroup$ It would be easier to understand if you wrote "the first [person] will e assigned to either group..." $\endgroup$
    – Michael
    May 29 '20 at 4:59
  • $\begingroup$ You can calculate $E[X_{n+1}^2|X_n]$. $\endgroup$
    – Michael
    May 29 '20 at 4:59
  • $\begingroup$ Thank you so much for the comment, if possible, could you be more specific $\endgroup$
    – koori
    May 29 '20 at 5:10
  • $\begingroup$ How did you calculate $E[X_n]$? I assume it was via $E[X_{n+1}|X_n]$ or $E[X_{n+1}|X_n=a]$. $\endgroup$
    – Michael
    May 29 '20 at 5:38
  • $\begingroup$ Thank you for the comments, I will try to use the conditional expectation to calculate $E(X_{n})$. Previously, I just assume that it is $n/2$ since the distribution is symmetric $\endgroup$
    – koori
    May 29 '20 at 5:54
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Similarly, we can calculate:
$E({X^{2}_{n+1}|X_{n}}) = (X_{n} +1)^{2}\frac{n-X_{n}}{n} +X_{n}^{2}\frac{X_{n}}{n} = (X_{n} +1)^{2} - \frac{2X_{n}+X_{n}^2}{n}$

$V({X_{n+1}|X_{n}}) = E({X^{2}_{n+1}|X_{n}}) - E({X_{n+1}|X_{n}})^2 =\frac{X_{n}}{n} - \frac{X_{n}^2}{n^2} $

$V({X_{n+1}}) = E_{X_n}(V({X_{n+1}|X_{n}})) + V_{X_n}(E({X_{n+1}|X_{n}})) = \frac{E(X_{n})}{n} - \frac{E(X_{n}^2)}{n^2} +(\frac{n-1}{n})^2V(X_n) $
Since we have $E(X_{n}) =\frac{n}{2} $ and $E(X_{n}^2) =V(X_n) + E(X_n)^2$, Finally we can get

$V(X_{n+1}) = \frac{n-2}{n} V(X_{n}) + \frac{1}{4}$

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