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GaltonWatsonModell

The Galton–Watson model, also known as the Galton–Watson branching process, is a discrete-time stochastic model for the evolution of a population in generations. It was originally developed to study the extinction of surnames and has since become a fundamental tool in probability theory.

In the standard formulation, one starts with Z0 = 1 ancestor. Each individual in generation n produces

Key properties include the extinction probability q, defined as P(Zn eventually drops to 0). q is the

Variants include multi-type Galton–Watson processes and continuous-time branching processes. Applications span biology, epidemiology, population genetics, computer

a
random
number
of
offspring
according
to
a
fixed,
independent
and
identical
distribution
{p_k},
where
p_k
=
P(X
=
k)
and
X
denotes
the
number
of
children
of
a
single
individual.
If
Zn
is
the
size
of
generation
n,
then
Zn+1
=
X1
+
X2
+
...
+
XZn,
where
the
Xi
are
i.i.d.
copies
of
X.
The
offspring
distribution
is
encapsulated
by
its
generating
function
G(s)
=
sum_k
p_k
s^k.
smallest
nonnegative
solution
of
q
=
G(q).
If
the
mean
m
=
E[X]
is
less
than
or
equal
to
1,
then
q
=
1
(extinction
occurs
with
probability
1).
If
m
>
1,
then
q
<
1,
and
there
is
a
positive
probability
of
survival,
with
Zn
typically
growing
roughly
like
m^n
on
the
event
of
non-extinction.
science
(random
trees),
and
nuclear
chain
reactions.
The
model
is
named
for
Francis
Galton
and
Henry
William
Watson,
who
studied
surname
extinction
in
the
19th
century.