Conditional expectations : practice question

[frenchmarmot]

Hello,

I've just started studying for FRM part 1 (May 2020). Started with the Quantitative Analysis chapters. Most of it has been fine but now I'm struggling to understand question 4.14 (and its answer) in the GARP book.

I got every question right before 4.14. Nevertheless, I don't understand why we need to calculate a non-normalized (which I calculated thinking this was the the only calculation to do ) AND then normalize it. What's the rational behind normalizing ?





Answers from the book ?



source : FRM exam part 1 2020, Quantitative Analysis, p58-60, (GARP)

Thanks in advance for your help.

 

[amit.m.sharma]

The question looks incomplete. For simplicity let’s say the question spelled it out that you have to find the distribution for X1 given the condition that X2 <=0. Because we are redefining the space from X2 taking any of the original values to only 0 or -1 we need to scale the corresponding X1 probabilities so that they add up to 100percent.

Take the following question as an example. A woman has two children. One of the children is a boy. What is the probability that the other child is a boy too? Assume there is equal probability of having a boy or girl.

There are four possibilities: GG, GB, BG and BB. Given that one of the children is a boy, our space contains only three options, GB, BG and BB. So, the probability of having two boys is 1/3. If we had no prior knowledge that one of the children is a boy then the probability of two boys would be 1/4. Hope you see how the three options got scaled from 1/4 to 1/3.

 

[frenchmarmot]

Thanks for your answer.
I think I get your explanation, makes sense to me.

PS: Indeed, end of the question was missing. Just added it.