Assume that conditioned on random variable X, $Y$ is a gaussian random variable as follows:
$\hspace{7cm}Y \sim \mathcal{N}(0,X+1).$
Here, $X$ is a discrete RV which takes the value $k \times \sigma^2$ with probability$\frac{\binom{N}{k}}{2^N-1-N}$ where $k \in \{2,3,...N\}$.
How can I compute the PDF of $Y$?
My attempt: $f_Y(y)=\large\sum_{k=2}^{N}\small f_Y(y|X=x)Pr(X=x)$. But, I am unable to simplify this summation. Can someone help me simplify this? Is there a closed-form expression for the PDF of $Y$?
If it is not directly possible, is there an approximate expression assuming $N >>1$.?