- The joint distribution of two Bernoulli random variables, X and Y, is determined
by the 2 by 2 matrix of probabilities, p(i,j) = P(X=i,Y=j) for i=0,1 and j=0,1.
Suppose the (prior) distribution of (p(0,0),p(0,1),p(1,0),p(1,1)) is uniform
on the set {p(0,0)>0, p(0,1)>0, p(1,0)>0, p(1,1)>0,
p(0,0)+p(0,1)+p(1,0)+p(1,1)=1}. (This is the Dirichlet distribution with parameters
1,1,1,1.) Find the conditional distribution of the p(i,j) given that X and Y are
independent (i.e. given P(X=i,Y=j) = P(X=i)P(Y=j)).
Need a hint?
- A fair coin is tossed until it comes up heads and the number, X, of tails
before the first heads appears is recorded. Then two slips of paper are taken
and the number 3 to the power X is written on one of the slips and 3 to the
power X+1 is written on the other. One of these two slips is chosen at random
(probability 1/2 each) and given to you. It will have one of the numbers
1, 3 ,9, 27, 81, etc. written on it. After observing this number, call it z, you
are given the option of exchanging your slip for the other, unseen slip. Then
you are paid in dollars the number on the slip you possess. For which values
of z would you make the exchange? For which values of z would you make the
exchange if you had to pay a 10% commission to the broker who effects the exchange?
Need a hint?
- Kriegspiel is the game of chess in which the players are not allowed to
see the moves of the opponent. Instead, there is a "referee" who keeps track
of the each move on his own private board and tells the players if an attempted
move is legal. If an attempted move is legal, the player must play it; otherwise,
the player may try again. In addition, the referee announces all captures and
whether the captured object is a piece or a pawn. In the RAND rules, the referee
also announces checks, and possible captures by a pawn.
The question is: Can a blind person, hearing only the announcements of the referee,
deduce a mate for one of the players? Need a
hint?
- Two positive integers are chosen. The sum is revealed to logician A, and
the sum of the squares is revealed to logician B. Both A and B are given this
information and the information contained in this sentence. The conversation
between A and B goes as follows: B starts.
B: "I can't tell what the two numbers are."
A: "I can't tell what the two numbers are."
B: "I can't tell what the two numbers are."
A: "I can't tell what the two numbers are."
B: "I can't tell what the two numbers are."
A: "I can't tell what the two numbers are."
B: "Now I can tell what the two numbers are."
What are the two numbers? Need a hint?
- The outcome of a certain experiment results in three (possibly dependent)
random variables, X, Y, Z, each taking values either zero or one. (So there
are 8 possible outcomes to the experiment.) Is it possible to set up such an
experiment so that
P(X = Y) + P(X = Z) + P(Y = Z) < 1 ?
Need a hint?
Thomas S. Ferguson
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