# Math Help - Continous random variables questions

1. ## Continous random variables questions

I have having difficulties with this question

I have attached my question.

any help is appreichated

thanks

2. Ever heard of the law of the total probability?
$\mathbb{P}(X+Y\leq a)=\mathbb{E}[\mathbb{P}(X\leq a- Y|Y)]$

3. could you please explain that law by applying to my question.

thanks

4. $\mathbb{P}(X\leq 1/2 -Y |Y)=[1/2-Y]^+$ where [x]^+ is the maximum between x and 0 (because if Y is greater than 1/2, we cannot have a negative probability, and the prob of X being less than a negative number is 0).

Now
$
\mathbb{E}[1/2-Y]^+=\int_0^1 \max\{1/2-y,0\}\mathbb{P}(Y\in dy)=\int_0^{1/2} (1/2-y) dy
$

you can complete the rest.