  We present a statistical analysis of the X-ray bursts observed from the 2002
June 18 outburst of the Anomalous X-ray Pulsar (AXP) 1E 2259+586, observed with
the Proportional Counter Array (PCA) aboard the Rossi X-ray Timing Explorer. We
show that the properties of these bursts are similar to those of Soft
Gamma-Repeaters (SGRs). The similarities we find are the burst durations follow
a log-normal distribution which peaks at 99 ms, the differential burst fluence
distribution is well described by a power law of index -1.7, the burst fluences
are positively correlated with the burst durations, the distribution of waiting
times is well described by a log-normal distribution of mean 47 s, and the
bursts are generally asymmetric with faster rise than fall times. However, we
find several quantitative differences between the AXP and SGR bursts.
Specifically, the AXP bursts we observed exhibit a wider range of durations,
the correlation between burst fluence and duration is flatter than for SGRs,
the observed AXP bursts are on average less energetic than observed SGR bursts,
and the more energetic AXP bursts have the hardest spectra -- the opposite of
what is seen for SGRs. We conclude that the bursts are sufficiently similar
that AXPs and SGRs can be considered united as a source class yet there are
some interesting differences that may help determine what physically
differentiates the two closely related manifestations of neutron stars.
