What is the probability that the determinant of a matrix of order 2 is positive ,whose elements are i.u. ran.var.in the interval(0,1)?I have been thinking about the problem for quite a while and it seems to me a problem on geometric probability albeit in four dimensions.For if the elements of the matrix are a,b,c,d(row-wise) the value of the determinant is ad-bc.the problem reduces to finding the probability P(ad-bc&gt;0) subject to given domain and distributation of a,b,c,d.In our problem the sample space is the 4D unit cube.If we take the axes as X,Y,Z,U THE REQUISITE PROBABILITY IS GIVEN BY THE FRACTION OF 4D VOLUME OF THE UNIT CUBE FOR WHICH XY-ZU&gt;0. I would love to see a solution along these goemetrical lines.

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What is the probability that the determinant of a matrix of order 2 is positive ,whose elements are i.u. ran.var.in the interval(0,1)?I have been thinking about the problem for quite a while and it seems to me a problem on geometric probability albeit in four dimensions.For if the elements of the matrix are a,b,c,d(row-wise) the value of the determinant is ad-bc.the problem reduces to finding the probability P(ad-bc&amp;gt;0) subject to given domain and distributation of a,b,c,d.In our problem the sample space is the 4D unit cube.If we take the axes as X,Y,Z,U THE REQUISITE PROBABILITY IS GIVEN BY THE FRACTION OF 4D VOLUME OF THE UNIT CUBE FOR WHICH XY-ZU&amp;gt;0. I would love to see a solution along these goemetrical lines.

Chaya Garza · Accepted Answer

Explanation:The probability that the determinant is 0 is 0. Thus by symmetry the required probability is       1    2

pleitatsj1 · Accepted Answer

Step 1The products ad and bc have identical distributions, and   a  d  −  b  c is the convolution of the density distributions of ad and   −  b  c, which is symmetrical and centered at zero. So the probability is 1/2.Step 2Edit: that under the hypothesis that a,b,c,d have the same density distribution.

What is the probability that the determinant of a matrix of order 2 is positive ,whose elements are i.u. ran.var.in the interval(0,1)?

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