Bernoulli
Funktione de probableso de mase
Funktione de akumulati distributione
Parametres
p
>
0
{\displaystyle p>0\,}
real )
q
≡
1
−
p
{\displaystyle q\equiv 1-p\,}
Suporto
k
=
{
0
,
1
}
{\displaystyle k=\{0,1\}\,}
Funktione de probableso de mase (fpm)
q
por
k
=
0
p
por
k
=
1
{\displaystyle {\begin{matrix}q&{\mbox{por }}k=0\\p~~&{\mbox{por }}k=1\end{matrix}}}
Funktione de akumulati distributione (fad)
0
por
k
<
0
q
por
0
<
k
<
1
1
por
k
>
1
{\displaystyle {\begin{matrix}0&{\mbox{por }}k<0\\q&{\mbox{por }}0<k<1\\1&{\mbox{por }}k>1\end{matrix}}}
Medivalore
p
{\displaystyle p\,}
Mediane N/A
Mode
max
(
p
,
q
)
{\displaystyle {\textrm {max}}(p,q)\,}
Variantia
p
q
{\displaystyle pq\,}
Nonsimetreso
q
−
p
p
q
{\displaystyle {\frac {q-p}{\sqrt {pq}}}}
Kurtose
6
p
2
−
6
p
+
1
p
(
1
−
p
)
{\displaystyle {\frac {6p^{2}-6p+1}{p(1-p)}}}
Entropie
−
q
ln
(
q
)
−
p
ln
(
p
)
{\displaystyle -q\ln(q)-p\ln(p)\,}
mgf
q
+
p
e
t
{\displaystyle q+pe^{t}\,}
Kar. funk.
q
+
p
e
i
t
{\displaystyle q+pe^{it}\,}
In probableso teorie e statistike , li Bernoulli distributione , nomat segun suisi sientiiste Jakob Bernoulli , es diskreti probableso distributione , kel have valore 1 kun probableso
p
{\displaystyle p}
q
=
1
−
p
{\displaystyle q=1-p}
X es hasardal variable kun disi distributione, nus have:
Pr
(
X
=
1
)
=
{\displaystyle \Pr(X=1)=}
1
−
Pr
(
X
=
0
)
=
p
.
{\displaystyle 1-\Pr(X=0)=p.\!}
Li probableso-mase funktione f de disi distributione es
f
(
k
;
p
)
=
{
p
si
k
=
1
,
1
−
p
si
k
=
0
,
0
altrim.
{\displaystyle f(k;p)=\left\{{\begin{matrix}p&{\mbox{si }}k=1,\\1-p&{\mbox{si }}k=0,\\0&{\mbox{altrim.}}\end{matrix}}\right.}
Li expektati valore de Bernoulli hasardal variable X es
E
(
X
)
=
p
{\displaystyle E\left(X\right)=p}
variantia es
var
(
X
)
=
p
(
1
−
p
)
.
{\displaystyle {\textrm {var}}\left(X\right)=p\left(1-p\right).\,}
Li kurtose vada a infiniteso kun alti e basi valores de p , ma kun
p
=
1
/
2
{\displaystyle p=1/2}
Li Bernoulli distributione es membre del exponential familie .
Si
X
1
,
…
,
X
n
{\displaystyle X_{1},\dots ,X_{n}}
p , tand
Y
=
∑
k
=
1
n
X
k
∼
B
i
n
o
m
i
a
l
(
n
,
p
)
{\displaystyle Y=\sum _{k=1}^{n}X_{k}\sim \mathrm {Binomial} (n,p)}
binomial distributione ).
