(solution) Winter 2016 Math 130B Homework Submit the starred problems during

(solution) Winter 2016 Math 130B Homework Submit the starred problems during

Can u solve the last problem in the attached files, problem 2? I don’t know how to approach that one. 

Winter 2016 Math 130B Homework
Submit the starred problems during the Thursday?s discussion corresponding to
the due date.
But you are strongly recommended to work on all of these problems It is only
by practicing that you will be able to succeed in this class. Every Friday, the solutions of all the problems of the previous week will be posted on the website. If you
need some help to solve these problems do not hesitate to mention it during the
discussion sessions or to contact me or your TA. Due Date
Thu, Jan. 14
Thu, Jan. 21 Sections
6.1, 6.2 Problems
6, 9? , 11, 14? , 16, 19, 20? , 23, 26, 27
Theoretical : 5? , 9, 11?
28? , 29, 30?
Theoretical : see below 6.3 Homework assignment due on Thursday, January 21
In addition to the two starred problems of the book (28 an 30), you have to solve and
submit the two following problems.
Problem 1 : Let X1 and X2 be two independent exponential random variables with
respective parameter ?1 and ?2 . We suppose that ?1 = ?2 . Let Z = X1 + X2 .
(a) Find the probability density function of Z.
(b) Compute P( Z ? 2).
Problem 2 : Particles are subject to collisions that cause them to split into two parts,
each part being a fraction of the parent. Suppose that this fraction is uniformly
distributed between 0 and 1. Following a single particle through several splittings
we obtain a fraction of the original particle
n Zn = X1 · X2 · · · Xn = ? Xi ,
i =1 where Xi is uniformly distributed over [0, 1] for all i ? 1, n . We suppose moreover
that X1 , · · · Xn are independent.
(a) Let i ? 1, n . Show that Yi = ? ln Xi is an exponential random variable.
Specify its parameter.
n (b) Let Sn = ? Yi . Find the probability density function of Sn . i =1 1 (c) Prove that Zn = e?Sn . Find the cumulative distribution function of Zn in terms
of the cumulative distribution function of Sn . Prove then that the probability density
function of Zn is given by (? ln z)n?1 for 0 < z < 1,
( n ? 1) !
f n (z) = 0
otherwise. 2