Let {$X_i$, $1{leq}i{leq}n$} be a sequence of i.i.d. sequence of positive random variables with common absolutely continuous cumulative distribution function F(x) and probability density function f(x) and $E(X^2)$ < ${infty}$. The random variables X + Y and $frac{(X-Y)^2}{(X+Y)^2}$ are independent if and only if X and Y have gamma distributions. In addition, the random variables $S_n$ and $frac{sum_{i=1}^{m}(X_i)^2}{(S_n)^2}$ with $S_n=sum_{i=1}^{n}X_i$ are independent for $1{leq}m$ < n if and only if $X_i$ has gamma distribution for $i=1,{cdots},n$.