Homework 0, due 9/15

This homework is for us to gauge your background.  It will not count
for grading, but please do your best.  If you claim to find all these
problems to be trivial, we'll take your word for it.

 1. Nowadays, when Alice receives an email message, the probability
    is 0.6 that it is spam.  Given a spam message, the conditional
    probability is 0.1 that it contains the word "mortgage".  Given
    a non-spam message, the conditional probability is 0.005 that it
    contains the word "mortgage".  Alice just received yet another
    message with the word "mortgage".  How likely is it spam?

 2. Let n be a positive integer.

    (a) I pick n real numbers, randomly and independently, from the
        uniform distribution between 0 and 1.  What is the probability
        that the resulting sequence is in non-descending order?

    (b) If the sequence is not non-descending, I uniformly randomly
        permute it until it becomes non-descending.  How many times do I
        expect to permute the sequence?

    (c) Do your answers to (a) and (b) change if I pick the n real
        numbers from not the *uniform* distribution between 0 and 1 but
        the *normal* distribution with mean 0 and standard deviation 1?
        What if I pick n *integers* from the *uniform* distribution
        between 0 and 1?

    You may enjoy how the Jargon File defines "bogo-sort":

    (var.: stupid-sort) The archetypical perversely awful algorithm
    (as opposed to {bubble sort}, which is merely the generic bad
    algorithm). Bogo-sort is equivalent to repeatedly throwing a deck
    of cards in the air, picking them up at random, and then testing
    whether they are in order. It serves as a sort of canonical example
    of awfulness. Looking at a program and seeing a dumb algorithm, one
    might say "Oh, I see, this program uses bogo-sort." Esp. appropriate
    for algorithms with factorial or super-exponential running time in
    the average case and probabilistically infinite worst-case running
    time. Compare {bogus}, {brute force}.

    A spectacular variant of bogo-sort has been proposed which has
    the interesting property that, if the Many Worlds interpretation
    of quantum mechanics is true, it can sort an arbitrarily large
    array in linear time. (In the Many-Worlds model, the result of any
    quantum action is to split the universe-before into a sheaf of
    universes-after, one for each possible way the state vector can
    collapse; in any one of the universes-after the result appears
    random.) The steps are: 1. Permute the array randomly using a
    quantum process, 2. If the array is not sorted, destroy the
    universe (checking that the list is sorted requires O(n) time).
    Implementation of step 2 is left as an exercise for the reader.

 3. The probability density function for the normal distribution with
    mean 0 and standard deviation 1 is

        p(x) = exp(-x^2/2)/sqrt(2pi).

    Suppose we have n real numbers x[1],...,x[n] that are independently
    sampled from some normal distribution.  What mean and standard
    deviation for the normal distribution would maximize the probability
    density of the samples x[1],...,x[n]?  Justify your answer.