<s>
The	O
Poisson	O
distribution	O
is	O
also	O
the	O
limit	B-Algorithm
of	O
a	O
binomial	O
distribution	O
,	O
for	O
which	O
the	O
probability	O
of	O
success	O
for	O
each	O
trial	O
equals	O
divided	O
by	O
the	O
number	O
of	O
trials	O
,	O
as	O
the	O
number	O
of	O
trials	O
approaches	O
infinity	O
(	O
see	O
Related	O
distributions	O
)	O
.	O
</s>
<s>
For	O
sufficiently	O
large	O
values	O
of	O
,	O
(	O
say	O
>1000	O
)	O
,	O
the	O
normal	O
distribution	O
with	O
mean	O
and	O
variance	O
(	O
standard	B-General_Concept
deviation	I-General_Concept
)	O
is	O
an	O
excellent	O
approximation	O
to	O
the	O
Poisson	O
distribution	O
.	O
</s>
<s>
If	O
is	O
greater	O
than	O
about	O
10	O
,	O
then	O
the	O
normal	O
distribution	O
is	O
a	O
good	O
approximation	O
if	O
an	O
appropriate	O
continuity	B-General_Concept
correction	I-General_Concept
is	O
performed	O
,	O
i.e.	O
,	O
if	O
,	O
where	O
x	O
is	O
a	O
non-negative	O
integer	O
,	O
is	O
replaced	O
by	O
.	O
</s>
<s>
Variance-stabilizing	B-General_Concept
transformation	I-General_Concept
:	O
If	O
then	O
and	O
Under	O
this	O
transformation	O
,	O
the	O
convergence	B-Algorithm
to	O
normality	O
(	O
as	O
increases	O
)	O
is	O
far	O
faster	O
than	O
the	O
untransformed	O
variable	O
.	O
</s>
<s>
Other	O
,	O
slightly	O
more	O
complicated	O
,	O
variance	B-General_Concept
stabilizing	I-General_Concept
transformations	I-General_Concept
are	O
available	O
,	O
one	O
of	O
which	O
is	O
Anscombe	O
transform	O
.	O
</s>
<s>
See	O
Data	B-General_Concept
transformation	I-General_Concept
(	O
statistics	O
)	O
for	O
more	O
general	O
uses	O
of	O
transformations	O
.	O
</s>
<s>
It	O
is	O
also	O
an	O
efficient	O
estimator	O
since	O
its	O
variance	O
achieves	O
the	O
Cramér	O
–	O
Rao	O
lower	O
bound	O
(	O
CRLB	O
)	O
.	O
</s>
<s>
This	O
interval	O
is	O
'	O
exact	B-General_Concept
 '	O
in	O
the	O
sense	O
that	O
its	O
coverage	O
probability	O
is	O
never	O
less	O
than	O
the	O
nominal	O
.	O
</s>
<s>
When	O
quantiles	O
of	O
the	O
gamma	O
distribution	O
are	O
not	O
available	O
,	O
an	O
accurate	O
approximation	O
to	O
this	O
exact	B-General_Concept
interval	O
has	O
been	O
proposed	O
(	O
based	O
on	O
the	O
Wilson	O
–	O
Hilferty	O
transformation	O
)	O
:	O
</s>
<s>
The	O
posterior	O
mean	O
E[]	O
approaches	O
the	O
maximum	O
likelihood	O
estimate	O
in	O
the	O
limit	B-Algorithm
as	O
which	O
follows	O
immediately	O
from	O
the	O
general	O
expression	O
of	O
the	O
mean	O
of	O
the	O
gamma	O
distribution	O
.	O
</s>
<s>
Astronomy	O
example	O
:	O
photons	B-Application
arriving	O
at	O
a	O
telescope	O
.	O
</s>
<s>
Optics	B-Algorithm
example	O
:	O
the	O
number	O
of	O
photons	B-Application
emitted	O
in	O
a	O
single	O
laser	O
pulse	O
.	O
</s>
<s>
This	O
is	O
a	O
major	O
vulnerability	O
to	O
most	O
Quantum	O
key	O
distribution	O
protocols	O
known	O
as	O
Photon	B-Application
Number	O
Splitting	O
(	O
PNS	O
)	O
.	O
</s>
<s>
Under	O
an	O
assumption	O
of	O
homogeneity	O
,	O
the	O
number	O
of	O
times	O
a	O
web	B-Application
server	I-Application
is	O
accessed	O
per	O
minute	O
.	O
</s>
<s>
The	O
arrival	O
of	O
photons	B-Application
on	O
a	O
pixel	O
circuit	O
at	O
a	O
given	O
illumination	O
and	O
over	O
a	O
given	O
time	O
period	O
.	O
</s>
<s>
The	O
targeting	O
of	O
V-1	O
flying	O
bombs	O
on	O
London	O
during	O
World	O
War	O
II	O
investigated	O
by	O
R	B-Language
.	O
D	O
.	O
Clarke	O
in	O
1946	O
.	O
</s>
<s>
Gallagher	O
showed	O
in	O
1976	O
that	O
the	O
counts	O
of	O
prime	O
numbers	O
in	O
short	O
intervals	O
obey	O
a	O
Poisson	O
distribution	O
provided	O
a	O
certain	O
version	O
of	O
the	O
unproved	O
prime	O
r-tuple	O
conjecture	O
of	O
Hardy-Littlewood	O
is	O
true	O
.	O
</s>
<s>
Therefore	O
,	O
we	O
take	O
the	O
limit	B-Algorithm
as	O
goes	O
to	O
infinity	O
.	O
</s>
<s>
In	O
this	O
case	O
the	O
binomial	O
distribution	O
converges	B-Algorithm
to	O
what	O
is	O
known	O
as	O
the	O
Poisson	O
distribution	O
by	O
the	O
Poisson	O
limit	B-Algorithm
theorem	O
.	O
</s>
<s>
The	O
word	O
law	O
is	O
sometimes	O
used	O
as	O
a	O
synonym	O
of	O
probability	O
distribution	O
,	O
and	O
convergence	B-Algorithm
in	O
law	O
means	O
convergence	B-Algorithm
in	O
distribution	O
.	O
</s>
<s>
In	O
a	O
Poisson	O
process	O
,	O
the	O
number	O
of	O
observed	O
occurrences	O
fluctuates	O
about	O
its	O
mean	O
with	O
a	O
standard	B-General_Concept
deviation	I-General_Concept
These	O
fluctuations	O
are	O
denoted	O
as	O
Poisson	O
noise	O
or	O
(	O
particularly	O
in	O
electronics	O
)	O
as	O
shot	O
noise	O
.	O
</s>
<s>
The	O
correlation	O
of	O
the	O
mean	O
and	O
standard	B-General_Concept
deviation	I-General_Concept
in	O
counting	O
independent	O
discrete	O
occurrences	O
is	O
useful	O
scientifically	O
.	O
</s>
<s>
The	O
natural	O
logarithm	O
of	O
the	O
Gamma	O
function	O
can	O
be	O
obtained	O
using	O
the	O
lgamma	O
function	O
in	O
the	O
C	B-Language
standard	O
library	O
(	O
C99	O
version	O
)	O
or	O
R	B-Language
,	O
the	O
gammaln	O
function	O
in	O
MATLAB	B-Language
or	O
SciPy	B-Application
,	O
or	O
the	O
log_gamma	O
function	O
in	O
Fortran	B-Application
2008	O
and	O
later	O
.	O
</s>
<s>
R	B-Language
:	O
function	O
dpois(x, lambda )	O
;	O
</s>
<s>
Excel	B-Application
:	O
function	O
POISSON( x, mean, cumulative )	O
,	O
with	O
a	O
flag	O
to	O
specify	O
the	O
cumulative	O
distribution	O
;	O
</s>
<s>
Mathematica	B-Language
:	O
univariate	O
Poisson	O
distribution	O
as	O
PoissonDistribution[],	O
bivariate	O
Poisson	O
distribution	O
as	O
MultivariatePoissonDistribution[ {	O
} ]	O
,	O
.	O
</s>
<s>
R	B-Language
:	O
function	O
rpois(n, lambda )	O
;	O
</s>
<s>
A	O
simple	O
algorithm	O
to	O
generate	O
random	O
Poisson-distributed	O
numbers	O
(	O
pseudo-random	B-Algorithm
number	I-Algorithm
sampling	I-Algorithm
)	O
has	O
been	O
given	O
by	O
Knuth	O
:	O
</s>
<s>
Other	O
solutions	O
for	O
large	O
values	O
of	O
include	O
rejection	B-Algorithm
sampling	I-Algorithm
and	O
using	O
Gaussian	O
approximation	O
.	O
</s>
<s>
Inverse	B-Algorithm
transform	I-Algorithm
sampling	I-Algorithm
is	O
simple	O
and	O
efficient	O
for	O
small	O
values	O
of	O
,	O
and	O
requires	O
only	O
one	O
uniform	O
random	O
number	O
u	O
per	O
sample	O
.	O
</s>
