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FORMULARIO DI STATISTICA DESCRITTIVA
d = xmax xmin k hi = fi di oppure hi = ni di di = xi xi 1
M e =
x n 2
+x n 2 + 2 se^ n^ e^ pari
x n+ 2
se n e dispari
M e = xM e 1 + (xM e xM e 1 )
0 ; 5 FM e 1 FM e FM e 1
Q 1 =
x n 4
+x n 4 + 2 se^ n^ e^ pari
x n+ 4
se n e dispari
Q 1 = xQ 1 1 + (xQ 1 xQ 1 1 )
0 ; 25 F(Q 1 1) F(Q 1 ) F(Q 1 1)
Q 3 =
x 3 n 4
+x 3 n 4 + 2 se^ n^ e^ pari
x 3(n+1) 4
se n e dispari
Q 3 = xQ 3 1 + (xQ 3 xQ 3 1 )
0 ; 75 F(Q 3 1) F(Q 3 ) F(Q 3 1)
x = x = M (x) = 1 n
Xn
i=
xini 2 = M
x 2
1 n
Xn
i=
x 2 i ni
2 x =^
1 n
Xn
i=
(xi x)
2 ni = M
x 2
(x)
2 x =
p ^2 x =
v u u t 1 n
Xn
i=
(xi x)
2 ni
Range (x) = x(n) x(1) IQR = Q 3 Q 1 CV = x jxj
G = 1
X^ k
i=
f (^) i^2 G^ = G k k 1
H =
X^ k
i=
j fi log (fi)j H^ = H log(k)
R =
nX 1
i=
(pi qi)
nX 1
i=
pi
R^ e = 1
nX 1
i=
(pi+1 pi) (qi+1 + qi)
AF =
1 n
Xn
i=
(xi x)^3 ni
^3 x^ =^
1 n
Xn
i=
xi x x
ni K = 1 n
Xn
i=
xi x x
ni 3
2
X^ k
i=
X^ h
j=
(nij bnij )^2 bnij
2
^2 n
xy = 1 n
Xn
i=
(xi x)
yi y
ni =
= M (xy) M (x) M (y)
xy =
xy
xy
1 n
Xn
i=
xiyi xy
v u u t 1 n
Xn
i=
(xi x)
2 ni
v u u t 1 n
Xn
i=
(yi y)
2 ni
ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó
by = b^0 + b^1 x b^1 =
Cov (x; y)
^2 x
b 0 =^ y^ ^
b 1 x
FORMULARIO DI STATISTICA INFERENZIALE
P (?) = 0
P
A
= 1 P (A)
P (A [ B) = P (A) + P (B) P (A \ B)
P (AjB) =
P (A\B) P (B) P^ (B)^ >^0 P^ (BjA) =^
P (B\A) P (A) P^ (A)^ >^0
P (A \ B) = P (B) P (AjB) = P (A) P (BjA) P (A \ B) = P (A) P (B)
P (HijE) =
P (Hi\E) P (E) =^
P (Hi)P (EjHi) X^ m
j=
P (Hj )P (EjHj )
r = E (X r ) =
P
x r i pi^
r = E (X r ) =
+R 1
x r f (x) dx
= E (X) =
P
xipi = E (X) =
+R 1
xf (x) dx
V AR (X) = E (X )
2
P
(xi )
2 pi = E
X
2
[E (X)]
2 V AR (X) =
+R 1
(xi )
2 f (x) dx
X U (a; b) P (X = x) = 1 b a (cost.) E (X) = b+a 2
V AR (X) =
(b a)^2 12
X Ber (p) P (X = x) = p x (1 p)
1 x E (X) = p V AR (X) = p (1 p)
X Bin (n ; p) P (X = x) =
n x
p x (1 p)
n x E (X) = np V AR (X) = np (1 p)
X Hg (N; S; n) P (X = x) =
0 @
S
x
1 A
0 @
N S
n x
1 A
0 @ N n
1 A
E (X) = np
p = S N
V AR (X) = np (1 p)
N n N 1
X P o () P (X = x) = x e x! E^ (X) =^ ^ V AR^ (X) =^
X Geo (p) P (X = x) = p (1 p)
x 1 E (X) = 1 p V AR^ (X) =^
1 p p^2
X N
1 ; ^21 =?
X N
2 ; ^22 =?
^21 = ^22 =?
P
X 1 X 2
tt (n 1 +n 2 ^2 ;^2 )
Sp
q 1 n 1
1 n 2
X 1 X 2
Sp
q 1 n 1
1 n 2
S
2 p =^
(n 1 1)S 12 +(n 2 1)S^22 n 1 +n 2 2
TEST DELLE IPOTESI
Z =
X p^ n
T =
X p^ s n
Z =
(X 1 X 2 ) ( 1 2 )
r ^21 n 1 +^
^22 n 2
T =
(X 1 X 2 ) ( 1 2 )
s (n 1 1)S^21 +(n 2 1)S^22 n 1 +n 2 2
1 n 1 +^
1 n 2
Z =
Sn q n^ p^0 p 0 (1 p 0 ) n
= N (0; 1) Z =
(^) Sn 1 n ^
Sn 2 n
(p 1 p 2 ) s Sn 1 n
1 Sn 1 n
n 1 +
Sn 2 n
1 Sn 2 n
n 2
(n 1)S^2 ^2
2 (n 1)
2
Pk
i=
(fi pi)
2
pi
Pk
i=
(ni npi)
2
npi
2
Pk
i=
P^ k
j=
(nij cij )
2
cij
Pk
i=
P^ k
j=
(nij bnij )
2
bnij
Disuguaglianza di Ceby sev: (^) P fjX j < "g 1 1 "^2 P fjX j < kg 1 ^2 k^2
REGRESSIONE
x = 1 n
Xn
i=
xi y = 1 n
Xn
i=
yi
Cov (X; Y ) =
P
xiyi (x y) s 2 x =
X^ n
i=
(xi x)^2 n 1 s
2 y =
X^ n
i=
(yi y)^2 n 1
y b = 0 +^
1 x^
1 =^
Cov(x;y) ^2 x
0 =^ y^ ^
1 x
s" =
q SSE n 2
r (n 1)s^2 y COV (X;Y ) s^2 x n 2
Statistica test : T =
1 ^1 s 1
s 1
p s" (n 1)s^2 x
Intervallo di conf idenza : 1 ^ t^2 ;(n 2)^ ^ s^ 1
R^2 = 1
PSSE (yi y)^2
