Let x and y b uniformly distributed, independent random variables on [0,1]. The probability that the distance between x and y is <1/2 is
A) 1/4
B) 1/3
C) 1/2
D) 2/3
E) 3/4
GR9768 Q 47

 Posts: 27
 Joined: Tue Apr 06, 2010 8:22 am
Re: GR9768 Q 47
that is easy, man, just draw a pic.
you want the following to be satisfied: xy<0.5
that becomes
xy < 0.5 and xy > 0.5
which is equivalent to
y > x0.5 and y < x + 0.5
now draw a unit square [0,1]x[0,1], abscissa represents variable X, and ordinate corresponds to variable Y. draw these two lines above, I mean, plot the inequalities. you'll get the region between two parallel lines, and its area is just what you need  that is the probability of xy<0.5. it's easy to figure out the area, I leave it up to you
you want the following to be satisfied: xy<0.5
that becomes
xy < 0.5 and xy > 0.5
which is equivalent to
y > x0.5 and y < x + 0.5
now draw a unit square [0,1]x[0,1], abscissa represents variable X, and ordinate corresponds to variable Y. draw these two lines above, I mean, plot the inequalities. you'll get the region between two parallel lines, and its area is just what you need  that is the probability of xy<0.5. it's easy to figure out the area, I leave it up to you