Tuesday, April 30, 2013

Distribution of The Difference of Two Uniform Distribution Variable

Suppose U1~uniform(0,1), U2~uniform(0,1), and U1 is independent of U2, then what is the distribution of U1-U2?

The solution is in the picture attached. Then based on this, what is the distribution of U1+U2? The density should be of the same shape while it moves 1 unit to the right. The reason is: if U2 is uniform, then U3=1-U2 is uniform. so U1+U2 is the same as U1+U2=U1-U3+1. So it's density is the same as U1-U2 with i unit right transfer.

The answer can be verified in R. Density plot in R:
# Z is the difference of the two uniform distributed variable
plot(density(z), main="Density Plot of the Difference of Two Uniform Variable", col=3)
# X is the summation of the two uniform distributed variable
plot(density(x), main="Density Plot of the Summation of Two Uniform Variable", col=3)
Inline image 1
Inline image 1

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