How to find the conditional PDF of Y which is a sum of the source X and noise Z.

Question

I met a question in my information theory test, in fact I think it is a probability problem. The question is:

Suppose that an analog communication channel with input $X$ and output $Y$ is given by $Y = X + Z$, where $Z$ is a Gaussian noise with mean $0$ and variance $\sigma^2>0$, that is, its probability density function is given by $$p_Z\left(z\right)=\frac1{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{z^2}{2\sigma^2}\right)$$ Further suppose that the input $X$ is also a Gaussian random variable with mean $0$ and variance $1$, and that $X$ and $Z$ are independent. Answer the following questions.
(1) Find the joint probability density function $p_{XZ}\left(x,z\right)$ of the pair of random variables $\left(X,Z\right)$.
(2) Find the conditional probability density function $p_{Y\vert X}\left(y\vert x\right)$ of Y under the condition that $X = x$.

For the first question, I know because X and Z are independent, so I could just write down the $p_X\left(x\right)$ like $p_Z\left(z\right)$ and then multiply them.
But I got stuck on question 2. Even I know $X = x$, for the ranges of $X$ and $Z$ are all $\left(-\infty,+\infty\right)$ (though probability is low), how will this help me understand $Y$ as I know $X = x$?
If the question asked me to calculate $p_Y\left(y\right)$, of course I know how to do it, for the sum of two normal distribution is also a normal distribution. Also, maybe because I do not know Q2, I cannot image what $P_{Y\vert X}\left(y\vert x\right)$ looks like...
Could you give me some hints on that? Thank you!

Answer 1

$$[Y\mid X=x]\stackrel{d}{=}[X+Z\mid X=x]\stackrel{d}{=}[x+Z\mid X=x]\stackrel{d}{=}x+Z$$

where the last statement is based on independence of $X$ and $Y$.

I used $\stackrel{d}{=}$ here because actually we are dealing with equality of distributions while "$Y$ under condition $X=x$" cannot be recognized as a random variable.

It is stated here that under condition $X=x$ the distribution of $Y$ coincides with the distribution of $x+Z$.