Distribuci´on binomial (demostraciones)
P(X=x) = n
xpx(1 −p)n−x, x = 0,1, . . . , n;MX(t) = pet+ 1 −pn, t ∈R.
•Media: E[X] = np
C´alculo a partir de la definici´on:
E[X] =
n
X
x=0
x n
x!px(1 −p)n−x=
n
X
x=1
x n
x!px(1 −p)n−x=1=np
n
X
x=1 n−1
x−1!px−1(1 −p)n−x=m=n−1
y=x−1
=np
m
X
y=0 m
y!py(1 −p)m−y=np.
C´alculo a partir de la funci´on generatriz de momentos:
E[X] = dMX(t)
dt t=0
=npet+ 1 −pn−1pett=0 =np.
•Momento de orden dos: E[X2] = n(n−1)p2+np
C´alculo a partir de la definici´on:E[X2] = E[X(X−1)] + E[X] = E[X(X−1)] + np
E[X(X−1)] =
n
X
x=0
x(x−1) n
x!px(1 −p)n−x=
n
X
x=2
x(x−1) n
x!px(1 −p)n−x=2
=n(n−1)p2
n
X
x=2 n−2
x−2!px−2(1 −p)n−x=m=n−2
y=x−2=n(n−1)p2
m
X
y=0 m
y!py(1 −p)m−y=
=n(n−1)p2.
C´alculo a partir de la funci´on generatriz de momentos:
E[X2] = d2MX(t)
dt2t=0
=nh(n−1) pet+ 1 −pn−2p2e2t+petpet+ 1 −pn−1it=0 =n(n−1)p2+np.
•Varianza: V ar[X] = E[X2]−(E[X])2=np(1 −p).
1x6= 0 ⇒x n
x!=n n−1
x−1!
2x6= 0,1⇒x(x−1) n
x!=n(n−1) n−2
x−2!
