Expected value and variance from random variable Given the following
$Y_1 \sim \mathcal{N}(μ, σ^2 )$
and
$Y_2=α+βY_1+U \;where \; Y_1 \;and \;U\;is\;independent\;and\;U∼\mathcal{N}(0,v^2)$
Let $μ=350$ and $σ^2 =12365$
How would i calculate the expected value and variance from $Y_2$ ?
And how can i find the distribution of it?
 A: Pretty much by definition of expected value: $$E(Y_2) = E(\alpha + \beta Y_1 + U)\\ = E(\alpha) + E(\beta Y_1)+ E(U)\\ = \alpha + \beta E(Y_1) + E(U)$$
And variance:
$$var(Y_2) = E(Y_2^2) - E(Y_2)^2$$
We need to find $E(Y_2^2)$, which is
$$E\left[(\alpha + \beta Y_1 + U)^2\right]\\
= E\left[\alpha^2 + \alpha \beta Y_1 + \alpha U\\ + \alpha \beta Y_1 + \beta^2 Y_1^2 + \beta Y_1 U\\ + \alpha U + \beta Y_1 U + U^2\right].$$
The rest you can probably do with linearity and some arithmetic, I hope this helps.
Edit: Most of the work comes from linearity of the expect value function.
A: Because of independence:
$m_Y=E(Y_2)=\alpha +\beta \mu$
$\sigma^2_Y=var(Y_2)=var(\alpha)+var(\beta Y_1)+var (U)$
$=\beta \sigma^2+ v^2$
$Y_2$ ~ $N(m_Y,\sigma^2_Y)$.
You need values for $\alpha$, $\beta$ and $v^2$.
A: You still haven't given values for $v, a, b,$
so I have supplied some.
The distribution of $Y_2$
is normal.
In my simulation in R below, I have used a million
iterations, so the sample mean and SD of $Y_2$
should agree with the respective population values
to several significant digits--I guess, good enough to
find mistakes in applying the formulas.
set.seed(2021)
y1 = rnorm(10^6, 350, 111.198)
u =  rnorm(10^6, 0, 50)
a = 10;  b = 50
y2 = a + b*y1 + u
summary(y2);  sd(y2);  var(y2)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  -8248   13765   17511   17513   21263   43899 
[1] 5566.788  # aprx SD(Y2)
[1] 30989134  # aprx Var(Y2)

Here are histograms of the simulated distributions
along with the corresponding normal density curves.

R code for figure.
par(mfrow=c(1,3))
hist(y1, prob=T, col="wheat")
 curve(dnorm(x,350,111.198), add=T, lwd=2)
hist(u, prob=T, col="wheat")
 curve(dnorm(x,0,50), add=T, lwd=2)
hist(y2, prob=T, col="skyblue2")
 curve(dnorm(x, 17813, 5566.788), add=T, lwd=2)
par(mfrow=c(1,1))


*

*You say in a comment that you don't follow how to
calculate some of the results for $Y_2.$
That is
not a comment that prompts specific explanations.


*I have had a quick look at answers by @herbsteinberg and @JLF, which seem fine:
(+1) for each.


*Can you say specifically
what you don't understand? If there are several
difficulties, you might start by saying what the first
couple of them are.
