[SOLVED] Probability & statistics for eecs homework 03

1. A system composed of 5 homogeneous devices is shown in the following figure. It is
said to be functional when there exists at least one end-to-end path that devices on
such path are all functional. For such a system, if each device, which is independent
of all other devices, functions with probability p, then what is the probability that the
system functions? Such a probability is also called the system reliability.
Figure 1: An illustration of the system composed of 5 homogeneous devices.2. (a) Suppose that in the population of gamers, being good at Genshin Impact is independent of being good at Apex (with respect to some measure of “good”). A
certain school club has a simple admission procedure: admit an applicant if and
only if the applicant is good at Genshin Impact or is good at Apex.Intuitive explain why it makes sense that among gamers that the club admits,
being good at Apex is negatively associated with being good at Genshin Impact,
i.e., conditioning on being good at Apex decreases the chance of being good at
Genshin Impact.(b) (Berkson’s paradox) Show that if A and B are independent and C = A ∪ B, then
A and B are conditionally dependent given C (as long as P(A ∩ B) > 0 and
P(A ∪ B) < 1), with
P(A | B ∩ C) < P(A | C).3. A fair die is rolled repeatedly, and a running total is kept (which is, at each time, the
total of all the rolls up until that time). Let pn be the probability that the running
total is ever exactly n (assume the die will always be rolled enough times so that the
running total will eventually exceed n, but it may or may not ever equal n).(a) Write down a recursive equation for pn. Your equation should be true for all
positive integers n, so give a definition of p0 and pk for k < 0 so that the recursive
equation is true for small values of n.(b) Find p7.
(c) Give an intuitive explanation for the fact that pn → 1/3.5 = 2/7 as n → ∞.4. There are n types of toys, which you are collecting one by one. Each time you buy a
toy, it is randomly determined which type it has, with equal probabilities. Let pi,j be
the probability that just after you have bought your ith toy, you have exactly j toy
types in your collection, for i ≥ 1 and 0 ≤ j ≤ n.(a) Find a recursive equation expressing pi,j in terms of pi−1,j and pi−1,j−1, for i ≥ 2
and 1 ≤ j ≤ n.
(b) Describe how the recursion from (a) can be used to calculate pi,j .5. Link is an immortal drunk man who wanders around randomly on the integers. He
starts at the origin, and at each step he moves 1 unit to the right or 1 unit to the left,
with probabilities p and q = 1 − p respectively, independently of all his previous steps.
Let Sn be his position after n steps.(a) Let pk be the probability that the drunk ever reaches the value k, for all k ≥ 0.
Write down a recursive equation for pk (you do not need to solve it for this part).
(b) Find pk, fully simplified; be sure to consider all 3 cases: p < 1/2, p = 1/2, and
p > 1/2.