Let suppose I have two conitnuous random variables $X$ and $Y$, which are dependent, and I want to find their joint CDF $P(X+Y \leq \alpha)$. Lets suppose I know their marginals $P(X)$ and $P(Y)$, but I don't know their joint $P(X,Y)$ or the conditional expansion $P(X|Y)P(Y)$ or $P(Y|X)P(X)$ because I don't know the conditionals.
Question:
Is there anyway I could find a lower bound on $P(X+Y \leq \alpha)$? What about an upper bound?
Context:
I have the following random variable: $z = a+\mathbf{c}^{T} \mathbf{w} + \mathbf{w}^{T}\mathbf{w}$, where $\mathbf{w}$ is a vector of i.i.d. standard Gaussian random variables, and $a,\mathbf{c}^{T}$ are deterministic constants. I found the pdf of $\mathbf{w}^{T}\mathbf{w}$, and the pdf of $\mathbf{c}^{T} \mathbf{w} + a$ is just a shifted and scaled standard Gaussian. I also have their characteristics functions. I want to find $P(z \geq \alpha)$. To do so, I can find $P(z \geq \alpha) = 1 - P(z \leq \alpha)$
So for me, $X = \mathbf{c}^{T} \mathbf{w} +a$ and $Y = \mathbf{w}^{T}\mathbf{w}$. Can I upper bound $P(z \leq \alpha)$ by assuming they are independent, finding the resulting pdf via their characteristic functions, and then computing the cdf via integration? If so, I could lower bound $P(z \geq \alpha)$.
In general, what type of bound (upper or lower) would such a procedure give me?