Suppose $\alpha_1, ..., \alpha_n $ are independent identically distributed random variables, $ a_1, ..., a_n,b_1,...,b_n $ are non-zero constants. Show that if $ \sum_{i=1}^{n}a_i\alpha_i $ and $\sum_{i=1}^{n}b_i\alpha_i$ are independent, then $\alpha_1, ..., \alpha_n $ are normal variables.

Suppose $\alpha_1, ..., \alpha_n $ are independent identically distributed random variables, $ a_1, ..., a_n,b_1,...,b_n $ are non-zero constants. Is it true that if $ \sum_{i=1}^{n}a_i\alpha_i $ and $\sum_{i=1}^{n}b_i\alpha_i$ are independent, then $\alpha_1, ..., \alpha_n $ are normal variables?

Suppose $\alpha_1, ..., \alpha_n $ are independent identically distributed random variables, $ a_1, ..., a_n,b_1,...,b_n $ are non-zero constants. Show that if $ \sum_{i=1}^{n}a_i\alpha_i $ and $\sum_{i=1}^{n}b_i\alpha_i$ are independent, then $\alpha_1, ..., \alpha_n $ are both normal variables.

Suppose $\alpha_1, ..., \alpha_n $ are independent identically distributed random variables, $ a_1, ..., a_n,b_1,...,b_n $ are non-zero constants. Show that if $ \sum_{i=1}^{n}a_i\alpha_i $ and $\sum_{i=1}^{n}b_i\alpha_i$ are independent, then $\alpha_1, ..., \alpha_n $ are normal variables.