For $n=1$ one can easily check that $2X_a+X_b$ has the binomial distribution on $\{0,1,2\}$ with success probability $\theta$. Then the expectation and the variance easily follow for any $n$. More precisely, one also deduces that $2X_a+X_b$ has the binomial distribution on $\{0,\ldots,2n\}$ with success probability $\theta$.