Write $\mathbb P(X+Y \leq c \mid X=d)$ in terms of $f$

Question

Problem: Let $X,Y$ random variables with joint pdf $f(x,y)$. Write $\mathbb P(X+Y \leq c \mid X=d)$ in terms of $f$.

Attempt: I have problems with this kind of things:

$$\mathbb P(X+Y \leq c \mid X=d) = \frac {\mathbb P (X+Y \leq c , X=d) }{ f_X(d)} = \frac {\int_{-\infty}^{c-d} f_Y(t) \, dt}{ f_X(d)}.$$

Why is it wrong?

Thanks!

Answer 1

Even simpler! $$\Bbb P(X+Y \leq c \mid X=d)=\Bbb P(d+Y \leq c )=\Bbb P(Y \leq c-d)=\int_{-\infty}^{c-d}\int_{-\infty}^\infty f(x,y) \, dx \, dy$$