I have not been able to show that the following two stochastic variables have the same distribution. My question is as follows:
Let $$ X_1, X_2,..., X_n $$ be independent and identically distributed. Furthermore, $$ a_1, a_2,...,a_n $$ are arbitrary real numbers. Set $$ Y_1 = a_1X_1 + a_2X_2 + .... +a_nX_n $$ $$ Y_2 = a_nX_1 + a_{n-1}X_2+....+a_1X_n $$
Show that $$y_1 \stackrel {d}{=} y_2 $$
I have tried proving this using the definition of the characteristic function $$ \varphi_{Y_1}(t) = E[e^{itY_1}]=E[e^{it(a_1X_1+a_2X_2+...+a_nX_n)}] $$ $$ \varphi_{Y_2}(t) = E[e^{itY_2}]=E[e^{it(a_nX_1 + a_{n-1}X_2+....+a_1X_n)}] $$
but I can't continue from here. Is this a correct approach to proving the above? Furthermore, the question says nothing about the quantity of a's.
(If we, for instance, would have known that n is large, we could have used the Central Limit Theorem which would say that the distribution of Xk approaches a normal distribution, and then used the characteristic function of a normal distribution to prove that Y1 have the same distribution as Y2. But there is no such information in the question. Do I have to assume this to be able to solve the question?)
Thank you! Jam
Hint: If $(X_k)_{1\leqslant k\leqslant n}$ is i.i.d. then $(Y_k)_{1\leqslant k\leqslant n}$ defined by $Y_k=X_{n+1-k}$ for every $1\leqslant k\leqslant n$ is distributed like $(X_k)_{1\leqslant k\leqslant n}$, in particular, $a_1X_1+a_2X_2+\cdots+a_nX_n$ and $a_1Y_1+a_2Y_2+\cdots+a_nY_n$ are identically distributed.