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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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Quote:
 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.

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