1)

Exercise 1* (Law of total probability) Let ((1, 29,13) be a discrete probability space and 31,... ,B'\" g Q be events with 13(31') > O (i=1,...,n) and where U B!- : e, (1) i=1 Bl, ..., B\" are pairwise disjoint. (2) Give a proof for the law of total probability, which states that for any event A g 9 PM) = iPcBMA | a). \"21 Exercise 2* (Unifom Distribution) Let X m (4'01, 5) be a continuous, uniformly distributed random variable on ((1,5). (a) Show that the expectation of X is E(X) 2 32%. (b) Show that the probability for X to fall into some sub-interval of ((1,3)) is proportional to the length of the sub-interval