A car company tests an upcoming car model for gas consumption $(x_1)$ and oil consumption $(x_2)$.From past tests it is known that the expected gas consumption is and the expected oil consumption is 1 Liter per 130kms(I believe these mean the Avg value).It is also known that the consumpion follows normal distribution and also the matrix of covariance is
$$ =\begin{pmatrix} 0.10 & 0.05 \\ 0.05 & 0.10 \end{pmatrix} $$
A new variable $ = \frac{1}{2} _ + \frac{1}{2}_$ represents the total consumption.
According to measurements, the gasoline consumption is 1.5 liters in 10,000 kilometers, and the engine oil consumption is 2 liters in 1,000 kilometers based on the measurements obtained from the tests.
$1)$What is the distribution of the total consumption?
$2)$What is the correlation between the two consumptions?
$3)$Assuming zero correlation between the two consumptions, calculate how the distribution of the total consumption changes.
I would like someone to guide me through an exercise like this so I can undestand the logic behind these problems and also correct me if I made any mistakes since i have answered questions 2 and 3.
Ps$(1)$: Since both $x_1$ and $x_2$ follow normal distribution we have that for $x_1$ $ N(\frac{8} {50},0.1)$ and for $x_2$ $ N(\frac{1}{130},0.1)$ In my experience that would mean that $y$ would also follow normal distribution but im not quite sure since we dont know if $x_1 , x_2$ are dependent or not.
Ps$(2)$: I believe that in the question number $2$ we are talking about Pearson correlation coefficient. So if thats the case i believe we have $r(x_1,x_2)= \frac{Cov(x_1,x_2)}{σ(x_1)*σ(x_2)}=\frac{0.05}{\sqrt(0.10)*\sqrt(0.10)}=\frac{0.05}{0.10}=\frac{1}{2}=0.5$
Ps$(3)$: Since we have independent $x_1 , x_2$ then $μ=\frac{1}{2}*(\frac{8}{50}+\frac{1}{130})=\frac{109}{1300}$ and $σ^2=\frac{1}{4}*(0.1+0.1)=0.05$. So y follows $N(\frac{109}{1300},0.05)$