How to calculate the expectation and variance of a complex probability distribution

Assuming that the continuous random variables X1 and X2 are independent of each other, and the variances exist, the probability densities of X1 and X2 are $f_{1}(x)$
and $f_{2}(x)$, the probability density of random variable $Y_{1}$ is $f_{Y_{1}}(y)=frac{1}{2}left[f_{1}(y)+f_{2 }(y)right]$, random variablemathrm${Y}_{2}=
frac{1}{2}left(X_{1}+X_{2}right)$. Which of the following statements is correct (the answer is D)?

$$begin{array}{c}
&(A)& E Y_{1}>E Y_{2}, D Y_{1}>D Y_{2}
&(B)& E Y_{1}=E Y_{2}, D Y_{1}=D Y_{2}
&(C)& E Y_{1}=E Y_{2}, D Y_{1}<D Y_{2}
&(D)& E Y_{1}=E Y_{2}, D Y_{1}>D Y_{2}
end{array}$$

When I use the normal distribution to verify the D option, the following code keeps running:

Y1 = ProbabilityDistribution[(1/
     2) (PDF[NormalDistribution[μ1, σ1], x] + 
     PDF[NormalDistribution[μ2, σ2], x]), {x, -Infinity, 
   Infinity}]


Expectation[Y1, Y1 [Distributed] Y1]
Variance[Y1]
Y2 = TransformedDistribution[
  1/2 (x1 + x2), {x1 [Distributed] 
    NormalDistribution[μ1, σ1], 
   x2 [Distributed] NormalDistribution[μ2, σ2]}]
Expectation[Y2, Y2 [Distributed] Y2]
Variance[Y2]

How can I improve the code to get the desired result?