Finding a critical region for a mixture Let $\ X_1 , X_2 $ be two iid random variables with normal N ( $\theta,1 $) distribution.  Further , consider bernoulli random variable V with P(V=1) = $ \frac{1}{4} $ and which is independent of $\ X_1 , X_2 $ .
Define $ \ X_3 $ as =
$\ X_1 $, if V=0
$\ X_2 $, if V=1
For testing hypothesis H0 : $ \theta =0 $ versus H1 : $ \theta =1 $
Reject H0 if $\frac{\ X_1 + X_2 +X_3 }{3} $ >C
Find C such that test size becomes 0.05 .
Now in this question I could understand only two things , one that V is an indicator variable and bernoulli distribution and second that it should be converted to Z before adding two critical region.
Other than that I dont know how to proceed , please help.
My attempt
$ \ X_3 = \ X_1*P(v=0) + \ X_2*P(v=1 ) $
$ E(\ X_3) = E(\ X_1*P(v=0)) + E(\ X_2*P(v=1 )) $
$ E(\ X_3) = (\theta *3/4) + (\theta*1/4) $
$ E(\ X_3) = \theta  $
Now $ Var (\ X_3) = Var(\ X_1 *3/4) +  var(\ X_2*1/4) $
$ Var (\ X_3) = \frac{9}{16} +  \frac{1}{16} $
$ Var (\ X_3) = \frac{10}{16}  $
$\frac{\ X_1 + X_2 +X_3 }{1} \ ~~~ N( 3\theta , ??) $
$\frac{\ X_1 + X_2 +X_3 }{3} \ ~~~ N( \theta , ??) $
Please help to proceed .

I think that,
$$
X_3 = \cases{X_1, V = 0\\X_2, V = 1}
$$
Therefore,
$$X_3 \sim N(\theta, 1)$$
Check if $X_3$ is independent of $X_1$ and $X_2$.
So what you have presented is the average of 3 iid observations of the variable $X \sim N(\theta, 1)$.
It is known that $\bar{x} \sim N(\theta, \frac{\sigma}{n})$. Which in this case $\bar{x} \sim N(\theta, \frac{1}{\sqrt{3}})$.
So I reject $H_0$ if $P(\bar{x} > c) = P(Z > \frac{c - \theta }{1/\sqrt{3}}) = P(Z > \sqrt{3}(c - \theta)) = 0.05$.
It follows then that
$$c - \theta = \frac{1.64}{\sqrt{3}} \Rightarrow c = \theta + \frac{1.64}{\sqrt{3}}$$
