Continous random variables questions

**cooltowns** · December 6th 2009, 06:05 AM

I have having difficulties with this question

I have attached my question.

any help is appreichated

thanks

**Focus** · December 6th 2009, 02:53 PM

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

**cooltowns** · December 7th 2009, 08:32 AM

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

thanks

**Focus** · December 8th 2009, 04:14 AM

$\mathbb{P}(X\leq 1/2 -Y |Y)=[1/2-Y]^+$ 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
$<br /> \mathbb{E}[1/2-Y]^+=\int_0^1 \max\{1/2-y,0\}\mathbb{P}(Y\in dy)=\int_0^{1/2} (1/2-y) dy<br />$
you can complete the rest.

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