Statistics question

2. A stick of length 1 [in meter) is broken at a uniformly random point X. Given that X = 3;, another breakpoint Y is chosen uniformly on the interval [0,1']. (a) (2 pts) Find the joint PDF of X and 1". Be sure to specify their joint support. Hint: We can denote the PDF of Unif(a,b) by f(22) = fqu), where IA is an indicator function that takes on 1 if A is the case and 0 otherwise. You can denote the support using this indicator function or in the way I did during the class. An important property of the indicator is that IAIB = {AHE- (b) (1 pt) Find the marginal PDF of X, and specify the name and parameter(s) of X 's distri- bution. You may already know this distribution intuitively from the given setting, but please make sure to conrm your intuition via the formal derivation. (c) (1 pt) Find the conditional PDF of Y given X = :c, and specify the name and parameter(s) of Y's distribution conditioning on X = :13. You may already know this distribution intuitively from the given setting, but please make sure to conrm your intuition via the formal derivation. (d) (3 pts) Find the marginal PDF of Y and its appropriate support. (No need to check validity.) (e) (3 pts) Find the conditional PDF of X given Y = y and its appropriate support. You may get the essential part of this conditional PDF easily, but please make sure you clearly specify the constant term as well (to make it in a complete PDF form]