How should I approach this Conditional Probability Problem?

Question

Can anyone give a hint on how to begin this problem?

Suppose $Y = X^2 + W$ where $W$ is Gaussian $N(0, 1)$ noise. Then derive an expression for $P(Y\mid X)$.

I know about Bayes' Rule but I'm not sure how that helps me since I would get this

$$P(Y\mid X) = \frac{P(X\mid Y)P(Y)}{P(X)}$$

Answer 1

Hint: If $Z=c^2+W$ and $W\sim N(0,1)$ what is the distribution of $Z$?

Then the expressions for $\mathsf P(Z\leq z)$, and $f_{Z}(z)$ are....

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Now if $Y=X^2+W$ and $W\sim N(0,1)$ what is the distribution of $Y\mid X$?

Then the expressions for $\mathsf P(Y\leq y\mid X)$, and $f_{Y\mid X}(y)$ are....