[SOLVED] Probability & statistics for eecs homework 04

1. Consider the original Monty Hall problem, except that Monty enjoys opening door 2
more than he enjoys opening door 3, and if he has a choice between opening these two
doors, he opens door 2 with probability p, where 1
2 ≤ p ≤ 1.To recap: there are three doors, behind one of which there is a car (which you want),
and behind the others there are goats (which you don’t want). Initially, all possibilities
are equally likely for where the car is. You choose a door, which for concreteness we
assume is door 1. Monty (knows which door has the car) then opens a door to reveal
a goat, and offers you the option of switching.(a) Find the unconditional probability that the strategy of always switching succeeds
(unconditional in the sense that we do not condition on which of doors 2 or 3
Monty opens).(b) Find the probability that the strategy of always switching succeeds, given that
Monty opens door 2 (assume we always choose door 1 first).
(c) Find the probability that the strategy of always switching succeeds, given that
Monty opens door 3 (assume we always choose door 1 first).2. (a) Is there a discrete distribution with support {1, 2, 3, . . . }, such that the value of
the PMF at n is proportional to 1/n?
(b) Is there a discrete distribution with support {1, 2, 3, . . . }, such that the value of
the PMF at n is proportional to 1/n2?3. Let X be a random day of the week, coded so that Monday is 1, Tuesday is 2, etc. (so
X takes values 1, 2, . . . , 7 with equal probabilities). Let Y be the next day after X.
Do X and Y have the same distribution? What is P(X < Y )?4. There are two coins, one with probability p1 of Heads and the other with probability
p2 of Heads. One of the coins is randomly chosen (with equal probabilities for the two
coins). It is then flipped n ≥ 2 times. Let X be the number of times it lands Heads.(a) Find the PMF of X.
(b) What is the distribution of X if p1 = p2?
(c) Give an intuitive explanation of why X is not Binomial for p1 ∕= p2.5. For x and y binary digits (0 or 1), let x
�y be 0 if x = y and 1 if x ∕= y (this operation
is called exclusive or (often abbreviated to XOR), or addition mod 2).(a) Let X ∼ Bern(p) and Y ∼ Bern(1/2), independently. What is the distribution of
X �Y ?(b) With notation as in sub-problem (a), is X �Y independent of X? Is X �Y
independent of Y ? Be sure to consider both the case p = 1/2 and the case p ∕= 1/2.(c) Let X1, . . . , Xn be i.i.d. (i.e., independent and identically distributed) Bern(1/2)
R.V.s. For each nonempty subset J of {1, 2, . . . , n}, let
YJ = �
j∈J
Xj .Show that YJ ∼ Bern(1/2) and that these 2n − 1 R.V.s are pairwise independent,
but not independent.