Question

Let Yi ? Unif(0, 1) for i = 1, 2. Find the pdf of U = Y1 + Y2 using the method of cdfs.

Answer 1

The pdf of Y1 and Y2 is

$f_{Y_{1}}(y_{1})=1$

$f_{Y_{2}}(y_{2})=1$

Let $U=Y_{1}+Y_{2}$

The range of U will be 0 to 2. Using convolution we have

$f_{U}(u)=\int_{-\infty}^{\infty}f_{Y_{1}}(u-y_{2})f_{Y_{2}}(y_{2})dy_{2}=\int_{-\infty}^{\infty}f_{Y_{1}}(u-y_{2})dy_{2}$

Since Y2 is defined between 0 and 1 so

$f_{U}(u)=\int_{0}^{1}f_{Y_{1}}(u-y_{2}) dy_{2}$

Now since Y1 is defined between 0 and 1 so we have $0\leq u-y_{2}\leq 1$ . Now $0\leq u-y_{2}\leq 1$ can be written as $u-1\leq y_{2}\leq u$ . When z is between 0 and 1 then

$f_{U}(u)=\int_{0}^{u} 1 dy_{2}=u$

And when u is between 1 and 2 then

$f_{U}(u)=\int_{u-1}^{1} dy_{2}=2-u$

So required pdf is

$f_{U}(u)=\left\{\begin{matrix} u, & 0\leq u\leq 1\\ 2-u,&1\leq u\leq 2 \\ 0,& otherwise \end{matrix}\right.$