Thread: Sum of 2 independent uniform r.v.

1. Sum of 2 independent uniform r.v.

If X and Y are independent r.v, both uniformly distributed on (0, 1).

Given that $\displaystyle f_{X+Y}(a)=\int_{-\infty}^{\infty}f_X(a-y)f_Y(y)dy$

we obtain $\displaystyle f_{X+Y}(a)=\int_{0}^{1}f_X(a-y)dy$

For $\displaystyle 0\leq a\leq 1$, we have $\displaystyle f_{X+Y}=\int_{0}^{a}dy=a$

For $\displaystyle 1<a<2$, we have $\displaystyle f_{X+Y}=\int_{a-1}^{1}dy=2-a$

How do I determine the two upper and lower bounds?
Such as for this case we are using: $\displaystyle \int_{0}^{a}$ and $\displaystyle \int_{a-1}^{1}$ ??

Another question:

Given joint density of X and Y:

$\displaystyle f(x,y)=\frac{1}{4}(y-x)e^{-y}$
$\displaystyle -y<x<y, 0<y<\infty$
In order to find the density function of X, I meet the same problem:

for x>0, $\displaystyle f_X(x)=\frac{1}{4}\int_{x}^\infty (y-x)e^{-y} dy$

for x<0, $\displaystyle f_X(x)=\frac{1}{4}\int_{-x}^\infty (y-x)e^{-y} dy$
Why the two different lower bounds?

2. Originally Posted by noob mathematician
If X and Y are independent r.v, both uniformly distributed on (0, 1).

Given that $\displaystyle f_{X+Y}(a)=\int_{-\infty}^{\infty}f_X(a-y)f_Y(y)dy$

we obtain $\displaystyle f_{X+Y}(a)=\int_{0}^{1}f_X(a-y)dy$

For $\displaystyle 0\leq a\leq 1$, we have $\displaystyle f_{X+Y}=\int_{0}^{a}dy=a$

For $\displaystyle 1<a<2$, we have $\displaystyle f_{X+Y}=\int_{a-1}^{1}dy=2-a$

How do I determine the two upper and lower bounds?
Such as for this case we are using: $\displaystyle \int_{0}^{a}$ and $\displaystyle \int_{a-1}^{1}$ ??

[snip]
Read example 7.3 p292 here: http://www.dartmouth.edu/~chance/tea...k/Chapter7.pdf