If we choose 2 numbers from [0,2], independent from each other,what is the probability of their geometric mean being higher than 1/6?

cortejosni 2022-08-16 Answered
Probability of geometric mean being higher than 1/6
If we choose 2 numbers from [0,2], independent from each other,what is the probability of their geometric mean being higher than 1/6?
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Answers (2)

Kyle George
Answered 2022-08-17 Author has 22 answers
Step 1
Note that X Y > 1 / 6 iff Y > 1 / ( 36 X ) where X, Y are i.i.d uniform [0,2] random variables. Since X,Y are i.i.d uniform [0,2] it follows that P ( ( X , Y ) A ) = m ( A [ 0 , 2 ] 2 ) 4 i.e. (X,Y) is uniformly distributed on [ 0 , 2 ] 2 (here m means area) and A R 2 ). Letting A = { ( x , y ) R 2 36 x y > 1 } we have that P ( X Y > 1 / 6 ) = P ( Y > 1 / ( 36 X ) ) = 1 4 0 2 1 / 36 x 2 d y d x which you can compute.
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janine83fz
Answered 2022-08-18 Author has 2 answers
Explanation:
The values of x and y, you are interested in, lie in a square [ 0 ; 2 ] × [ 0 ; 2 ]. So you have a chart of y = 1 36 x , and an answer to your problem equals to an area of intersection of an epigraph and a given square divided by an area of a square. Which equals to 4 1 / 72 2 1 36 x 2 72 4
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