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?

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.
