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 October 12th, 2018, 06:18 AM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 592 Thanks: 87 Geometric Probability Probability with two variables x,y If $\displaystyle x\in [0,1] \;$ and $\displaystyle \; y\in [0,1]$ Find probability such that $\displaystyle x+y \leq 1$ Question is why we use division of area created by line $\displaystyle f(t)=t-1$ and total area ? Post solution with explanation October 12th, 2018, 09:02 AM #2 Senior Member   Joined: Sep 2015 From: USA Posts: 2,529 Thanks: 1389 Draw the line from the point (1,0) to the point (0,1). This is the line described by $y = 1-x$ This line divides the positive unit square into 2 triangles. Ordered pairs in the lower triangle (including the line) are such that $x + y \leq 1$ so $P[x+y \leq 1] = \dfrac{\text{area of lower triangle}}{\text{area of entire positive unit square}} = \dfrac{\frac 1 2 }{1} = \dfrac 1 2$ October 12th, 2018, 11:54 AM #3 Senior Member   Joined: Oct 2013 From: New York, USA Posts: 660 Thanks: 87 Couldn't it be said that x + y is uniformly distributed from 0 to 2 with 0.5 probability of being between 0 and 1 and 0.5 probability of being between 1 and 2? October 12th, 2018, 01:05 PM   #4
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 Originally Posted by EvanJ Couldn't it be said that x + y is uniformly distributed from 0 to 2 with 0.5 probability of being between 0 and 1 and 0.5 probability of being between 1 and 2?
The joint probability density of (X,Y) is 2D uniform.

Their sum won't be 1D uniform. Work it out and see. October 13th, 2018, 08:25 AM #5 Senior Member   Joined: Oct 2013 From: New York, USA Posts: 660 Thanks: 87 I don't know what 2D and 1D mean, but I figured it out. The numbers closest to 1 will be the most frequent sums. It's like how the sum of two dice has a mean of 7, with 7 as the the mode and the probability decreasing to the extremes of 2 and 12. October 13th, 2018, 09:52 AM   #6
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Quote:
 Originally Posted by EvanJ I don't know what 2D and 1D mean
2 dimensional and 1 dimensional.

(X,Y) has a 2 dimensional joint distribution

X+Y has a 1 dimensional distribution. Tags geometric, probability Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post USAMO Reaper Probability and Statistics 8 February 2nd, 2015 02:05 PM USAMO Reaper Geometry 1 February 2nd, 2015 09:00 AM yotastrejos Advanced Statistics 3 November 13th, 2013 03:12 AM hellxfire Algebra 2 May 27th, 2009 01:33 PM lprox015 Algebra 1 May 7th, 2007 08:45 AM

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