SOLUTION TO PROBLEM 31 OF CHAPTER 3
We are assuming that Xand Yare independent random variables both having a
discrete uniform density on the first Npositive integers. So, in particular,
fX(x)=
1
N,x=1,2,...,N;
0,otherwise.
and similarly for fY.
We want to determine the density of X+Y. For convenience, set Z=X+Y.
By (25) on page 72,
fZ(z)=
z
X
x=0
fX(x)fY(z−x).(∗)
Note that the range of Zis {2,3,..., 2N}.
Now, a summand in (∗) is nonzero if and only if
1≤x≤N
1≤z−x≤Nor 1≤x≤N
z−N≤x≤z−1
or
max{1,z−N}≤x≤min{N,z −1}.
But,
max{1,z−N}=(1,z≤N;
z−N, z ≥N+1
and
min{N,z −1}=(z−1,z≤N;
N, z ≥N+1
Thus, if z=2,3,...,N,
f
Z(z)=
z−1
X
x=1
1
N·1
N=z−1
N2,
if z=N+1, N+2,...,2N,
f
Z
(z)=
N
X
x=z−N
1
N·1
N=2N+1−z
N2,
and fZ(z) = 0 otherwise.