Download If X ~ Uni(a,b) then

Formulas
Statistics
Chapter 4
General Addition rule:
P(A or B) = P(A) + P(B) –
P(A and B)
Special multi. rule (for indep.
events, P(A|B) = P(A))
P(A) P(B) = P(A and B)
Chapter 5
Chapter 6
E(X) =  =  xi P(X=xi)
Var(X) = 2 =  (x1-)2P(X=xi)
Prob(a<X<b) = ab p(x) dx
E(X) =  = ab xp(x) dx
Var(X) =  = ab (x-)2p(x) dx
If X ~ Bin(n, )
P(X=x) = nCx x (1-)n-x
E(X)
= n
Var(X) = n (1-)
Special Add. rule (for exclusive
events, i.e. P(A and B) = 0)
P(A or B) = P(A) + P(B)
Permutation
nPr = the number of ways to put
numbers 1 to r in n boxes
Complement Events (A happen
= B does not happen)
combination
nCr = the number of ways to put
If X ~ Geo(p)
P(X=x) =  (1-)x-1
If X ~ Exp()
P(A) = 1 – P(B)
r ticks in n boxes
E(X)
Var(X)
P(X<x) = 1 – e-x
E(X) = 1/
Var(X) = 1/2
Definition of conditional
probability
P(A|B) = P(A and B) / P(B)
= 1/
= (1-)/ 2
If X ~ Poi()
P(X=x) = (x e- ) / (x!)
E(X)
= Var(X)
=
General multiplication rule
P(A|B) P(B) = P(A and B)
6
If X ~ Uni(a,b)
P(c<X<d) = (d-c) / (b-a)
E(X) = (a+b) / 2
Var(X) = (b-a)2 / 12
Formulas
Statistics
Chapter 7
Chapter 8
Population Sample
Size

n
Mean

Variance

X
s
Random?
No
yes
Known?
No
yes
Want?
Yes
no
If n is large enough (n30) or
the underlying distribution is
normal, then
Z=
t=
X
/ n
X
s/ n
~ N(0, 1)
If n is large enough (n30) or
the underlying distribution is
normal, then
Limits of C.I. = X  z / 2
Limits of C.I. =
X  t / 2
2

s
n
P(1  P)
n
P

n
n
N n
N 1
N n
N 1
The min. sample size necessary
to find C.I.
z  
n =   /2 
 E 
2
z 
n   (1   )  / 2 
 E 
If the population size (N) is
finite (i.e. n/N > 0.05)
C.I. = X  z / 2
n
n
If n is large enough
C.I. = P  z / 2
s
C.I. = P  z P(1  P)
~ t(n-1),
If n is large enough (both n and
n(1-) > 5) we can assume
Z=
C.I. =
X  t / 2
N n
N 1
(1  ) ~ N(0,1)
n
7
2
Formulas
Statistics
Chapter 9
Chapter 10
If n is large enough (n30) or
the underlying distribution is
normal, then you can use the
If both samples are large enough,
or (both of the underlying
distributions
are
normal,
following statistics
independent and having the
same sd.) then you can use the
following statistics:
Z=
X  0
/ n
Chapter 11
If n1 and n2 are large enough
then you can use the following
statistics
P1  P2
Z=
Pc (1  Pc ) Pc (1  Pc )

n1
n2
X1  X 2
t=
Z=
X  0
s/ n
12
n1

 22
PC =
n2
X1  X 2
n1  n 2
The coefficient of correlation
(X  X)(Y  Y)
s x s y (n  1)
r=
Under some assumptions, you
can use the following statistics
r n2
t=
If n is large enough (both n and
n(1-) > 5) then you can use the
following statistics
Z=
X1  X 2
t=
P  0
0 (1  0 )
n
If n is large enough or d follows
normal distribution then you can
use the following statistics for
the difference between pairs of
data
 1
1 

s 2p 


 n1 n 2 
(df = n1+n2-2)
sp 2 =
(n1  1)s12  (n 2  1)s 22
n1  n 2  2
t=
d
sd / n
(df = n-1)
8
1 r2
To find the regression equation
b=
r
sy
s x , a = Y  bX