Sufficiency and completeness in two distribution

Question

I have two variables $X_1$ and $X_2$ which follows exponential and double exponential with rate $\theta$

$X_1\sim\frac1\theta \exp((-x/\theta)),\quad X>0$

$X_2\sim\frac2\theta \exp(-2x/\theta)),\quad X>0$

$X_1$ and $X_2$ are independent. Then

How can we prove $X_1 + 2X_2$ is sufficient?

And how to check that $X_1 + X_2$ is sufficient?

$X_2$ is sufficient for $f_2$ but I am confused

Answer 1

$X_1+2X_2$ is sufficient by the factorization theorem. It is also a minimal sufficient statistic. Therefore, $X_1+X-2$ is not sufficient because it is not a function of the minimal sufficient statistic. You can also prove it is not sufficient directly using the same idea described here.