The range of non-identically distributed binomial variables Lets say we have three independent variables:
$$\eqalign{
X_{1}\sim &B(n,\frac{1}{2}+\beta) \cr 
X_{2}\sim &B(n,\frac{1}{2}) \cr
X_{3}\sim &B(n,\frac{1}{2}-\beta).
}$$
I'm looking for the distribution of 
$$\max(X_{1},X_{2},X_{3})-\min(X_{1},X_{2},X_{3}).$$
For sufficiently large $n$ we have 
$$\Pr(\max(X_{1},X_{2},X_{3})=k)\approx \Pr(X_{1}=k)$$
and similarly 
$$\Pr(\min(X_{1},X_{2},X_{3})=k)\approx \Pr(X_{3}=k).$$
So using those two we have:  
$$\eqalign{
&\Pr(\max(X_{1},X_{2},X_{3})-\min(X_{1},X_{2},X_{3})=k) \cr
&\approx \Pr(X_{1}-X_{3}=k) \cr
&=(\frac{1}{2}+\beta)^k  (\frac{1}{2}-\beta)^{2n+k}\sum\limits_{i=k}^{n}\binom{n}{i}\binom{n}{i-k} \left(\frac{1+2\beta}{1-2\beta} \right)^{2i}
}$$
but this doesn't seem to be easy to approximate...  
I was thinking about another approach. Binomial variables can be approximated by normal distributions and the sum of normally distributed variables is well known, so it would be easy to calculate. 
The question: Are there any inequalities that could give a better approximation? I'm aware only of the Chernoff bound but it is not very helpful here.
 A: For $n \le 20$ just compute the distribution by brute force.
For example, with $n=3$ the pdf is
$$\begin{array}{}
 k=0: &-\frac{7}{64} \left(-1+4 \beta ^2\right)^3 \cr
 k=1: &\frac{39}{256} \left(1-4 \beta ^2\right)^2 \left(3+4 \beta ^2\right) \cr 
 k=2: &-\frac{3}{128} \left(-15-92 \beta ^2+560 \beta ^4+192 \beta ^6\right) \cr 
 k=3: &\frac{1}{256} \left(21+564 \beta ^2+1392 \beta ^4+448 \beta ^6\right)
\end{array}$$
This was produced by the Mathematica code
Clear[f, i, q];
f[n_, p_, k_] := Binomial[n, k] p^k (1 - p)^(n - k);
i[k1_, k2_, k3_, k_] := If[Max[{k1, k2, k3}] - Min[{k1, k2, k3}] == k, 1, 0];
q[n_, k_, p_] := 
  Sum[f[n, 1/2 + p, k1] f[n, 1/2, k2] f[n, 1/2 - p, k3] 
     i[k1, k2, k3, k], {k1, 0, n}, {k2, Max[0, k1 - k], Min[n, k1 + k]}, 
     {k3, Max[0, k1 - k, k2 - k], Min[n, k1 + k, k2 + k]}];
With[{n=3}, Table[q[n, k, \[Beta]], {k,0,n}]]

Calculations of the entire distribution for $n=20$ take four seconds.  Here are plots for $\beta = 0, 1/6, 1/3$.  (As $\beta$ increases, the distribution shifts to the right.)

A: Call $R_n=\max(X_1,X_2,X_3)-\min(X_1,X_2,X_3)$. The usual LLN shows that $R_n/n$ converges to $2\beta$ almost surely and in $L^1$. (The distribution of $R_n$ for a given fixed $n$ is an altogether different story.)
