A rectangular array of mn numbers arranged in n rows, each consisting of m columns, is said to contain a saddlepoint if there is a number that is both the minimum of its row and the maximum of its column. For instance, in the array 1 3 2 0 − 2 6 .5 12 3 the number 1 in the first row, first column is a saddlepoint. The existence of a saddlepoint is of significance in the theory of games. Consider a rectangular array of numbers as described previously and suppose that there are two individuals— A and B—who are playing the following game: 4 is to choose one of the numbers 1, 2,. .., n and B one of the numbers 1, 2,. . ., m. These choices are announced simultaneously, and if A chose i and B chose j. then A wins from B the amount specified by the number in the i th row, j th column of the array. Now suppose that the array contains a saddle point—say the number in the row r and column k call this number x r k . Now if player A chooses row r, then that player can guarantee herself a win of at least x r k (since x r k is the minimum number in the row r). On the other hand, if player B chooses column k, then he can guarantee that he will lose no more than x r k (since x r k is the maximum number in the column k). Hence, as A has a way of playing that guarantees her a win of x r k and as B has a way of playing that guarantees he will lose no more than x r k it seems reasonable to take these two strategies as being optimal and declare that the value of the game to player A is x r k . If the nm numbers in the rectangular array described are independently chosen from an arbitrary continuous distribution, what is the probability that the resulting array will contain a saddlepoint?
A rectangular array of mn numbers arranged in n rows, each consisting of m columns, is said to contain a saddlepoint if there is a number that is both the minimum of its row and the maximum of its column. For instance, in the array 1 3 2 0 − 2 6 .5 12 3 the number 1 in the first row, first column is a saddlepoint. The existence of a saddlepoint is of significance in the theory of games. Consider a rectangular array of numbers as described previously and suppose that there are two individuals— A and B—who are playing the following game: 4 is to choose one of the numbers 1, 2,. .., n and B one of the numbers 1, 2,. . ., m. These choices are announced simultaneously, and if A chose i and B chose j. then A wins from B the amount specified by the number in the i th row, j th column of the array. Now suppose that the array contains a saddle point—say the number in the row r and column k call this number x r k . Now if player A chooses row r, then that player can guarantee herself a win of at least x r k (since x r k is the minimum number in the row r). On the other hand, if player B chooses column k, then he can guarantee that he will lose no more than x r k (since x r k is the maximum number in the column k). Hence, as A has a way of playing that guarantees her a win of x r k and as B has a way of playing that guarantees he will lose no more than x r k it seems reasonable to take these two strategies as being optimal and declare that the value of the game to player A is x r k . If the nm numbers in the rectangular array described are independently chosen from an arbitrary continuous distribution, what is the probability that the resulting array will contain a saddlepoint?
Solution Summary: The author explains how the probability of a saddle point in an array of size n is calculated by using the following expression:
A rectangular array of mn numbers arranged in n rows, each consisting of m columns, is said to contain a saddlepoint if there is a number that is both the minimum of its row and the maximum of its column. For instance, in the array
1
3
2
0
−
2
6
.5
12
3
the number 1 in the first row, first column is a saddlepoint. The existence of a saddlepoint is of significance in the theory of games. Consider a rectangular array of numbers as described previously and suppose that there are two individuals— A and B—who are playing the following game: 4 is to choose one of the numbers 1, 2,. .., n and B one of the numbers 1, 2,. . ., m. These choices are announced simultaneously, and if A chose i and B chose j. then A wins from B the amount specified by the number in the
ith row, jth column of the array. Now suppose that the array contains a saddle point—say the number in the row r and column k call this number
x
r
k
. Now if player A chooses row r, then that player can guarantee herself a win of at least
x
r
k
(since
x
r
k
is the minimum number in the row r). On the other hand, if player B chooses column k, then he can guarantee that he will lose no more than
x
r
k
(since
x
r
k
is the maximum number in the column k). Hence, as A has a way of playing that guarantees her a win of
x
r
k
and as B has a way of playing that guarantees he will lose no more than
x
r
k
it seems reasonable to take these two strategies as being optimal and declare that the value of the game to player A is
x
r
k
. If the nm numbers in the rectangular array described are independently chosen from an arbitrary continuous distribution, what is the probability that the resulting array will contain a saddlepoint?
Q1. A group of five applicants for a pair of identical jobs consists of three men and two
women. The employer is to select two of the five applicants for the jobs. Let S
denote the set of all possible outcomes for the employer's selection. Let A denote
the subset of outcomes corresponding to the selection of two men and B the subset
corresponding to the selection of at least one woman. List the outcomes in A, B,
AUB, AN B, and An B. (Denote the different men and women by M₁, M2, M3
and W₁, W2, respectively.)
Q3 (8 points)
Q3. A survey classified a large number of adults according to whether they were diag-
nosed as needing eyeglasses to correct their reading vision and whether they use
eyeglasses when reading. The proportions falling into the four resulting categories
are given in the following table:
Use Eyeglasses for Reading
Needs glasses Yes
No
Yes
0.44
0.14
No
0.02
0.40
If a single adult is selected from the large group, find the probabilities of the events
defined below. The adult
(a) needs glasses.
(b) needs glasses but does not use them.
(c) uses glasses whether the glasses are needed or not.
4. (i) Let a discrete sample space be given by
N = {W1, W2, W3, W4},
and let a probability measure P on be given by
P(w1) = 0.2, P(w2) = 0.2, P(w3) = 0.5, P(wa) = 0.1.
Consider the random variables X1, X2 → R defined by
X₁(w1) = 1, X₁(w2) = 2,
X2(w1) = 2, X2 (w2) = 2,
Find the joint distribution of X1, X2.
(ii)
X1(W3) = 1, X₁(w4) = 1,
X2(W3) = 1, X2(w4) = 2.
[4 Marks]
Let Y, Z be random variables on a probability space (, F, P).
Let the random vector (Y, Z) take on values in the set [0, 1] x [0,2] and let the
joint distribution of Y, Z on [0, 1] x [0,2] be given by
1
dPy,z (y, z) ==(y²z+yz2) dy dz.
harks 12 Find the distribution Py of the random variable Y.
[8 Marks]
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