24-08-2012, 02:02 PM
Sums of Independent Random Variables
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Sums of Discrete Random Variables
In this chapter we turn to the important question of determining the distribution of
a sum of independent random variables in terms of the distributions of the individual
constituents. In this section we consider only sums of discrete random variables,
reserving the case of continuous random variables for the next section.
We consider here only random variables whose values are integers. Their distribution
functions are then de¯ned on these integers. We shall ¯nd it convenient to
assume here that these distribution functions are de¯ned for all integers, by de¯ning
them to be 0 where they are not otherwise de¯ned.
Convolutions
Suppose X and Y are two independent discrete random variables with distribution
functions m1(x) and m2(x). Let Z = X + Y . We would like to determine the distribution
function m3(x) of Z. To do this, it is enough to determine the probability
that Z takes on the value z, where z is an arbitrary integer. Suppose that X = k,
where k is some integer. Then Z = z if and only if Y = z ¡ k. So the event Z = z