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990
10
583.7
170.1
http://www.toeic.or.jp/toeic/about/data/data_avelist/data_ave01_04.html
http://www.toeic.or.jp/toeic/ab...
990
10
583.7
170.1
http://www.toeic.or.jp/toeic/about/data/data_avelist/data_ave01_04.html
http://www.toeic.or.jp/toeic/ab...
D = {x1, x2, · · · , xn}
¯x =
1
n
nX
i=1
xi
2
=
1
n
nX
i=1
(xi ¯x)2
=
v
u
u
t 1
n
nX
i=1
(xi ¯x)2
=
1
n
nX
i=1
|xi ¯x|
=
1
n
nX
i=1
(xi ¯x)2
p
=
v
u
u
t 1
N
NX
i=1
(xi ¯x)2
p
probability
! 2 ⌦ = {!1, !2, · · · , !m}
⌦ = { , }
! 2 { , }
!(1)
= !(2)
=
!(n)
=
⌦ = {1, 2, 3, 4, 5, 6}
!(1)
= !(2)
=
!(n)
=
⌦ = {!1, !2, · · · , !49870000}
!(1)
= !43890298 = 171cm
!(2)
= !29184638 = 16...
!(1)
= !(2)
=
!(n)
=!(3)
=
!1 !2 !3 !4 !5 !6 !7 !8 !9 !10
= {!1, !2, !3, · · · , !10}
! 2 ⌦ = {ID1, ID2, ID3, · · · , ID10}
⌦ !
!
X = X(!)
⌦ !
!
X(!1) = 0
X(!2) = 0
X(!3) = 0
X(!4) = 0
X(!5) = 0
X(!6) = 0
X(!7) = 0
X(!8) = 0
X(!9) = 0
X(!10) = 100
!
{! 2 ⌦ : X(!) 2 A}
{X 2 A}
X(!) X
{! 2 ⌦ : X(!) 2 A}
!1 !2 !3 !4 !5 !6 !7 !8 !9 !10
A X(!) = 100Ac
X(!) = 0
!5 or !9
PX (A) = P(X 2 A) = P({! 2 ⌦ : X(!) 2 A})
⌦
!5, !9 !5, !9
PX (A) =
#({! 2 ⌦ : X(!) 2 A})
#( )
=
#(!5, !9)
#( )
=
2
10
= 0.2
PX(⌦) = 1
A1, A2, · · ·
PX ([iAi) =
X
i
PX (Ai)
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A12
0  PX(A)  1
X = X(!)
⌦
A
A
!1
!2
!3
!4
!5
!6
!7
!8
!11
!10
!9
!12
!13
!14
!15
!16
B
C
D
X(!) = 0
X(!) = 0
#A = #{! 2 ⌦ : X(!) = 0} = 7...
⌦
A
A
!1
!2
!3
!4
!5
!6
!7
!8
!11
!10
!9
!12
!13
!14
!15
!16
B
C
DX(!) = 0
P(X = 0) = PX(A) =
#{! 2 ⌦ : X(!) = 0}
#⌦
=
7
1...
{x1, x2, · · · , xk}
P(X = xi) = f(xi)
F(x) = P(X  x)
P(x < X  x + x)
x + xx
x x ! 0
f(x) = lim
x!0
P(x < X  x + x)
x
x + xx
f(x)
F(x) = P(X  x) =
Z x
1
f(u)du
f(a < x < b) =
Z b
a
f(x)dx
http://www.math.wm.edu/~leemis/2008amstat.pdf
P(X = x) = px
(1 p)1 x
(x = 0, 1)
#
#
p = 0.7
trial_size = 10000
set.seed(71)
#
data <- rbern(trial_size, p)
#
dens <- data.frame(y=c((1-p),p)*trial_size, x...
(x = 0, 1, · · · , n)
#
p = 0.7
trial_size = 10000
sample_size = 30
set.seed(71)
#
gen_binom_var <- function() {
return(sum(rbern(sample_size, p...
P(X = x) =
e x
x!
trial_size = 5000; width <- 1;
#
p = 0.7; n = 10;
np <- p*n
# n!∞ p!0 np=
n = 100000; p <- np/n
#
gen_binom_var <- functio...
f(x) =
1
p
2⇡ 2
exp
⇢
1
2
(x µ)2
2
( 1 < x < 1)
#
n <- 10000; p <- 0.7;
trial_size = 10000
width=10
#
gen_binom_var <- function() {
return(sum(rbern(n, p)))
}
result <- r...
( 1 < x < 1)
f(x) =
1
p
2⇡
exp
⇢
1
2
x2
#
n <- 10000; p <- 0.7
trial_size = 30000
width=0.18
#
gen_binom_var <- function() {
return(sum(rbern(n, p)))
}
result <- ...
f(x, k) =
(1/2)k/2
(k/2)
xk/2 1
e x/2
(0  x)
Xi
Z = X2
1 + · · · + X2
k
#
p <- 0.7; n <- 1000;
trial_size <- 100000; width <- 0.3;
df <- 3
# (3 )
gen_binom_var <- function() {
return(sum(rbern(n...
f(x, ) =
⇢
e x
(x 0)
0 (x < 0)
trial_size = 7000; width <- .01;
#
p = 0.7; n = 10; np <- p*n;
# n!∞ p!0 np=
n = 10000; p <- np/n
#
gen_exp_var <- functio...
f(x, ↵, ) =
↵
(↵)
x↵ 1
exp( x)
(0  x < 1)
↵X
i=1
Xi ⇠ (↵, )Xi ⇠ Exp( )
trial_size = 7000; width <- .035;
#
p = 0.7; n = 10; np <- p*n;
# n!∞ p!0 np=
n = 10000; p <- np/n; alpha <- 5
#
get_inter...
f(x, ↵, ) =
↵
(↵)
x (↵+1)
exp
✓
x
◆
(0  x < 1)
Xi ⇠ Exp( ) Z =
↵X
i=1
Xi ⇠ (↵, )
1/Z ⇠ IG(↵, )
trial_size = 7000; width <- .;
#
p = 0.7; n = 10; np <- p*n;
# n!∞ p!0 np=
n = 10000; p <- np/n; alpha <- 5
#
get_interval...
f(x) =
⇢
1 (0  x  1)
0 (otherwise)
Z = x1(1/2)1
+ x2(1/2)2
+ · · · + xq(1/2)q
width <- 0.02
p <- 0.5;
sample_size <- 1000
trial_size <- 100000
gen_unif_rand <- function() {
# sample_size 2
#
return (s...
f(x, a, b) =
⇢
(b a) 1
(a  x  b)
0 (otherwise)
a <- 5
b <- 8;
width <- 0.05
p <- 0.5
sample_size <- 1000
trial_size <- 500000
gen_unif_rand <- function() {
# sample_size...
f(x, ↵, ) =
1
B(↵, )
x↵ 1
(1 x) 1
(0 < x < 1)
Xi ⇠ U(0, 1)iid
(i = 1, 2, · · · , ↵ + 1)
width <- 0.03; p <- 0.5
digits_length <- 30; set_size <- 3
trial_size <- 30000
gen_unif_rand <- function() {
# digits_leng...
E[X] = X( )P( ) + X( )P( )
= 0 ⇥ 0.8 + 1, 000, 000 ⇥ 0.2
= 200, 000
E[X] =
X
x
xp(x)
µ
✓
n
x
◆
=
n!
(n x)!x!
E[X] =
nX
x=0
xP(x) =
nX
x=0
x
✓
n
x
◆
px
(1 p)n x
=
nX
x=0
x
n!
(n x)!x!
px
(1 p)n x
=
nX
x=0
n
(n ...
Var[X] = E[(X E[X])2
]
=
X
x
(x E[x])2
P(x)
= 2
µ
Var[x] = E[(X E[X])2
]
=
Z 1
1
(x E[x])2
f(x)dx
= 2
E[X] =
Z 1
1
xf(x)dx
= µ
E[g(X)] =
Z 1
1
g(x)f(x)dx
g(X) = (X E[X])2
E[ · ] =
Z 1
1
· f(x)dx
g(x) = xk
E[g(X)] = E[Xk
] =
Z 1
1
xk
f(x)dx
µ0
k
g(x) = (x E[x])k
E[g(X)] = E[(X E[X]])k
] =
Z 1
1
(x E[x])k
f(x)dx
µk
E[cX] = cE[X]
* E[cX] =
Z 1
1
cxf(x)dx = c
Z 1
1
xf(x)dx
= cE[X]
Var[cX] = c2
Var[X]
* Var[cX] =
Z 1
1
(cx E[cx])2
f(x)dx
=
Z 1
1
(cx cµ)2
f(x)dx
=
Z 1
1
c2
(x µ)2
f(x)dx
= c2
Z 1
1
(x µ)...
P(x < X 5 x + x, y < Y 5 y + y)
x, y ! 0
f(x, y) = lim
x, y!0
P(x < X 5 x + x, y < Y 5 y + y)
f(x, y)
g(x) =
Z 1
1
f(x, y)dy
h(y) =
Z 1
1
f(x, y)dx
g(x)
h(y)
EX,Y [ g(X, Y )] =
Z 1
1
Z 1
1
g(x, y)f(x, y)dxdy
g(x, y) = x0.8
y0.8 (x, y) ⇠ N((4, 4), S) S =

1 0.5
0.4 1
EX,Y [ g(X, ...
g(X, Y ) = (X µX)(Y µY )
Cov[X, Y ] = E[(X µX)(Y µY )]
g(X, Y ) = (X µX)(Y µY )
µX µX
µX µX
µY
µY
µY
µY
S1 = S2 =
S3 = S4 =

1 0.8
0.8 1

1 0.8
0.8 1

1 0
0 1

1 0.999
0.999...
f(x, y)
f(x, y) = g(x)h(y)
f(x, y) = g(x)h(y)
= 0
(x1, x2, · · · , xn)
x1
f(x1) =
Z
· · ·
Z
f(x1, · · · , xn)dx2 · · · dxn
x1
f(x1, · · · , xn) = f(x1) · · · f(xn)
x1 · · ·...
x1 · · · xn
g1(x1), · · · , gn(xn) x1 · · · xn
E[
nY
i=1
gi(xi)] =
nY
i=1
E[gi(xi)]
E[g1(x1)] E[gn(xn)]
E[
nY
i=1
gi(xi)] ...
x1 · · · xn
xi µi 2
i i = 1, 2, · · · , n
c = (c1, · · · , cn) c1x1 + · · · + cnxn
c1µ1 + · · · + cnµn
c2
1
2
1 + · · · + ...
E[c1x1 + · · · + cnxn]
=
Z 1
1
· · ·
Z 1
1
(c1x1 + · · · + cnxn)f(x1 · · · , xn)dx1 · · · dxn
= c1
Z 1
1
· · ·
Z 1
1
x1f(x...
Var[c1x1 + · · · + cnxn]
= E[{(c1x1 + · · · + cnxn) E[c1x1 + · · · + cnxn]}2
]
= E[{c1(x1 µ1) + · · · + c1(x1 µ1)}2
]
= E[...
x1 · · · xn
x1 · · · xn
xi
µ 2
(µ, 2
)
x1 · · · xn
T = x1 + · · · + xn
E[T] = E[x1 + · · · + xn]
= E[x1] + · · · + E[xn]
= nµ
Var[T] = Var[x1 + · · · + xn]
= Var...
¯x =
1
n
nX
i=1
xi =
1
n
T
E[¯x] =
1
n
E[T] = n ·
1
n
µ = µ
Var[¯x] = Var[
1
n
T] =
1
n2
Var[T] =
2
n
µ
2
Var[¯x] =
2
n
=
0.0833
500
= 0.000166
E[¯x] = 0.5
Var[¯x]
µ 2
P(|x µ| > ) 5
1
2
µ 2
1/ 2
= 1 ) P(|x µ| > ) 5 1
= 2 ) P(|x µ| > ) 5 1/4
= 3 ) P(|x µ| > ) 5 1/9
2
=
Z 1
1
(x µ)2
f(x)dx
=
Z
I1
(x µ)2
f(x)dx +
Z
I2
(x µ)2
f(x)dx +
Z
I3
(x µ)2
f(x)dx
2
=
Z
I1
(x µ)2
f(x)dx +
Z
I3
(x µ)...
x1 · · · xn µ
2
" > 0
lim
n!1
P{|¯xn µ| = "} = 0
¯xn =
1
n
nX
i=1
xi
¯xn µ
¯xn ! µ in P
" > 0
P(|¯xn µ| > ")
= P(|¯xn µ| > "
p
n
p
n
)
5
2
"2n
= 2
¯x=
=
1
2
f(x) =
1
p
2⇡ 2
exp
✓
(x µ)2
2 2
◆
f(x) =
1
p
2⇡
exp
✓
x2
2
◆
1 < x < 1
1 < x < 1
f(y) = y2
f(x) = x2
f(y) = y2
f(y) = exp( y2
)
z =
p
2y
f(z) = exp
✓
1
2
z2
◆
Z 1
1
e y2
dy =
p
⇡
Z 1
1
exp
✓
z2
2
◆
dz =
p
2⇡
Z 1
1
1
p
2⇡
exp
✓
z2
2
◆
dz = 1
dz =
p
2dy
Z 1
1
1
p
2⇡
exp
✓
z2
2
◆
dz
z =
x µ dz
dx
=
1
f(x) =
Z 1
1
1
p
2⇡ 2
exp
✓
(x µ)2
2 2
◆
dx
1/
D = (x1, · · · , xn) µ 2
¯x µ
/
p
n
, n ! 1 N(0, 1)
= 0.1, µ =
1
= 10, 2
=
1
2
= 100 ¯x = p
n
=
r
1
2n
=
r
1
0.01 ⇥ 10000
...
g(x) = ext
E[ext
] =
Z 1
1
ext
f(x)dx
Mx(t) = E[ext
]
Mx(t)
My(t)
x
t = 0
y
g(x) = ext
ext
= 1 + xt +
t2
2!
x2
+ · · · +
tk
k!
xk
+ · · ·
Mx(t) = E[ext
]
= E[1 + xt +
t2
2!
x2
+ · · · +
tk
k!
xk
+ ·...
Mx(t)
d
dtk
Mx(t) = E[xk
ext
]
t = 0
d
dtk
Mx(0) = E[xk
] = µ0
k
x ⇠ N(µ, )
Mx(t) = E[ext
] =
Z 1
1
ext 1
p
2⇡ 2
exp
✓
1
2
(x µ)2
2
◆
dx
z =
x µ
x = µ + z dx = dz
Mx(t) =
Z 1
1
e(µ+ z)t 1
p
2⇡ 2
exp
✓
1
2
z2
◆
dz
= eµt
Z 1
1
1
p
2⇡
exp
✓
tz
1
2
z2
◆
dz
= eµt
Z 1
1
1
p
2⇡
exp
✓
1
2
[z2...
(f · g)0
= f0
· g + f · g0
(f g)0
(x) = f0
(g(x))g0
(x)
M0
x(t) = (µ + 2
t)eµt+
2t2
2
M00
x (t) = (µ + 2
t)2
⇣
eµt+
2t2
2
...
Var[x] = E[x2
] (E[x])2
= (µ2
+ 2
) (µ)2
= 2
Var[x] = E[(x E[x])2
]
= E[x2
2E[x]x + E[x]2
)
= E[x2
] 2E[x]2
+ E[x]2
= E[x2...
D = (x1, · · · , xn) µ 2
¯x µ
/
p
n
, n ! 1
N(0, 1)
T = x1 + · · · + xn
T nµ
p
n
2T0
=
T nµ
p
n
=
¯x µ
1/
p
n
Mx(t)
My(t)
x
t = 0
y
T T0
=
T nµ
p
n
N(0, 2
)
Mxi
(t) = 1 + µ0
1t + µ0
2
t2
2!
+ µ0
3
t3
3!
+ · · ·
Mxi µ(t) = 1 + µ1t + µ2
t2
2!
+ µ3
t3
3!
+ · · ·
= 1 + 0 + 2 t2
2!
+...
xi µ
p
n
xi µ
p
n
Mxi µ
p
n
(t) = E[e
xi µ
p
n
t
]
= 1 + 2 t2
2!n
+ µ3
t3
3!n3/2
+ · · · + µk
tk
k!nk/2
+ · · ·
= 1 +
2
t2...
T0
=
x1 µ
p
n
+
x2 nµ
p
n
+ · · · +
xn µ
p
n
=
nX
i=1
xi µ
p
n
MT 0 (t) = MPn
i=1
⇣
xi µ
p
n
⌘(t) = E[e
Pn
i=1
⇣
xi µ
p
n
...
n ! 1
lim
n!1
MT 0 = lim
n!1
✓
1 +
1
n
2
t2
+ n
2
◆n
= e
2t2
2
lim
n!1
n = 0
N(0, 2
)
T0
=
T nµ
p
n
2
n = 100000
sample_size = 1000
rvs_list = []
m_list = []
for i in range(n):
unif_rvs = st.uniform.rvs(4.5, size=sample_size...
#
n = 10000
sample_size = 1000
rvs_list = []
m_list = []
m_unif = st.uniform.rvs(4, 2, size=sample_size)
m_beta_a = st.uni...
n_jobs = 20
n_each = int(n/n_jobs)
jobs = [Process(target=create_rvs, args=(n_each,)) for _ in range(n_jobs)]
manager = Ma...
D = (x1, · · · , xn)
✓0 = ˆ✓(X1, · · · , Xn)
ˆ✓lower(X1, · · · , Xn) 5 ✓0 5 ˆ✓upper(X1, · · · , Xn)
ˆ✓(X)
E[(ˆ✓(X) ✓)2
]
E[(ˆ✓(X) ✓)2
]
= E[{(E[ˆ✓(X)] ✓) + (ˆ✓(X) E[ˆ✓(X)])}2
]
= E[(E[ˆ✓(X)] ✓)2
+ 2(E[ˆ✓(X)] ✓)(ˆ✓(X) E[ˆ✓(X)]) + (ˆ✓(X) E[ˆ✓(X)...
E[¯x] =
1
n
E[T] = n ·
1
n
µ = µ
¯x
s2
=
1
n 1
nX
i=1
(xi ¯x)2
lim
n!1
P{|¯xn µ| = "} = 0 ¯xn ! µ in P
ˆ✓n(X) n ! 1
ˆ✓n(X) ! ✓ in P
ˆ✓n(X)
¯xn µ
Var[ˆ✓(X)]
ˆ✓(X)
D = (x1, · · · , xn) xi
f(xi)
nY
i=1
f(xi)
nY
i=1
f(xi|✓)
xi
`(✓|x1, x2, · · · , xn) =
nY
i=1
f(xi|✓)
x1, x2, · · · , x10
f(x1, x2, · · · , x10|µ, 2
) =
10Y
i=1
1
p
2⇡ 2
exp
✓
1
2
(xi µ)2
2
◆
`(µ, 2
|x1, x2, · · · , x10) =
10Y
i=1
1
p
2⇡ 2
exp
✓
1
2
(xi µ)2
2
◆
✓⇤
= arg max
✓
`(✓|x1, x2, · · · , xn)
log `(✓|x1, · · · , xn) ⌘ L(✓|x1, · · · , xn)
`
µ, 2
L(µ, 2
|x1, x2, · · · , x10) =
n
2
(2⇡)
n
2
log 2 1
2 2
nX
i=1
(xi µ)2
@L
@µ
=
1
2 2
nX
i=1
(xi µ)2
)
nX
i=1
xi = nµ
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数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
数学カフェ 確率・統計・機械学習回 「速習 確率・統計」
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数学カフェ 確率・統計・機械学習回 「速習 確率・統計」

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Veröffentlicht am

2017/4/22(土)【第18回数学カフェ】【確率・統計・機械学習回】
https://connpass.com/event/54301/
「速習 確率・統計」

本講演では、主に確率、統計の初心者の方や、プログラマ、エンジニアの方でデータ分析に興味を持っている方が確率統計のエッセンスを数理的に理解できることを目標に、データの集約方法から、大数の法則や中心極限定理など、確率・統計で利用される非常に重要な数学の定理などを紹介します。

Veröffentlicht in: Daten & Analysen
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数学カフェ 確率・統計・機械学習回 「速習 確率・統計」

  1. 1. Twitter
  2. 2. 990 10 583.7 170.1 http://www.toeic.or.jp/toeic/about/data/data_avelist/data_ave01_04.html http://www.toeic.or.jp/toeic/about/data/data_avelist/data_dist01_04.html
  3. 3. 990 10 583.7 170.1 http://www.toeic.or.jp/toeic/about/data/data_avelist/data_ave01_04.html http://www.toeic.or.jp/toeic/about/data/data_avelist/data_dist01_04.html
  4. 4. D = {x1, x2, · · · , xn} ¯x = 1 n nX i=1 xi 2 = 1 n nX i=1 (xi ¯x)2 = v u u t 1 n nX i=1 (xi ¯x)2
  5. 5. = 1 n nX i=1 |xi ¯x|
  6. 6. = 1 n nX i=1 (xi ¯x)2
  7. 7. p
  8. 8. = v u u t 1 N NX i=1 (xi ¯x)2 p
  9. 9. probability
  10. 10. ! 2 ⌦ = {!1, !2, · · · , !m} ⌦ = { , } ! 2 { , } !(1) = !(2) = !(n) =
  11. 11. ⌦ = {1, 2, 3, 4, 5, 6} !(1) = !(2) = !(n) = ⌦ = {!1, !2, · · · , !49870000} !(1) = !43890298 = 171cm !(2) = !29184638 = 168cm !(n) = !51398579 = 174cm
  12. 12. !(1) = !(2) = !(n) =!(3) = !1 !2 !3 !4 !5 !6 !7 !8 !9 !10 = {!1, !2, !3, · · · , !10} ! 2 ⌦ = {ID1, ID2, ID3, · · · , ID10}
  13. 13. ⌦ ! !
  14. 14. X = X(!) ⌦ ! ! X(!1) = 0 X(!2) = 0 X(!3) = 0 X(!4) = 0 X(!5) = 0 X(!6) = 0 X(!7) = 0 X(!8) = 0 X(!9) = 0 X(!10) = 100
  15. 15. ! {! 2 ⌦ : X(!) 2 A} {X 2 A} X(!) X
  16. 16. {! 2 ⌦ : X(!) 2 A} !1 !2 !3 !4 !5 !6 !7 !8 !9 !10 A X(!) = 100Ac X(!) = 0 !5 or !9
  17. 17. PX (A) = P(X 2 A) = P({! 2 ⌦ : X(!) 2 A}) ⌦ !5, !9 !5, !9 PX (A) = #({! 2 ⌦ : X(!) 2 A}) #( ) = #(!5, !9) #( ) = 2 10 = 0.2
  18. 18. PX(⌦) = 1 A1, A2, · · · PX ([iAi) = X i PX (Ai) A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 0  PX(A)  1
  19. 19. X = X(!) ⌦ A A !1 !2 !3 !4 !5 !6 !7 !8 !11 !10 !9 !12 !13 !14 !15 !16 B C D X(!) = 0 X(!) = 0 #A = #{! 2 ⌦ : X(!) = 0} = 7 #B = #{! 2 ⌦ : X(!) = 1} = 2 #C = #{! 2 ⌦ : X(!) = 2} = 4 #D = #{! 2 ⌦ : X(!) = 3} = 3
  20. 20. ⌦ A A !1 !2 !3 !4 !5 !6 !7 !8 !11 !10 !9 !12 !13 !14 !15 !16 B C DX(!) = 0 P(X = 0) = PX(A) = #{! 2 ⌦ : X(!) = 0} #⌦ = 7 16 P(X = 1) = PX (B) = #{! 2 ⌦ : X(!) = 1} #⌦ = 2 16 P(X = 2) = PX(C) = #{! 2 ⌦ : X(!) = 2} #⌦ = 4 16 P(X = 3) = PX(D) = #{! 2 ⌦ : X(!) = 3} #⌦ = 3 16
  21. 21. {x1, x2, · · · , xk} P(X = xi) = f(xi) F(x) = P(X  x)
  22. 22. P(x < X  x + x) x + xx x x ! 0 f(x) = lim x!0 P(x < X  x + x) x
  23. 23. x + xx f(x) F(x) = P(X  x) = Z x 1 f(u)du f(a < x < b) = Z b a f(x)dx
  24. 24. http://www.math.wm.edu/~leemis/2008amstat.pdf
  25. 25. P(X = x) = px (1 p)1 x (x = 0, 1)
  26. 26. # # p = 0.7 trial_size = 10000 set.seed(71) # data <- rbern(trial_size, p) # dens <- data.frame(y=c((1-p),p)*trial_size, x=c(0, 1)) # ggplot() + layer(data=data.frame(x=data), mapping=aes(x=x), geom="bar", stat="bin", bandwidth=0.1 ) + layer(data=dens, mapping=aes(x=x, y=y), geom="bar", stat="identity", width=0.05, fill="#777799", alpha=0.7)
  27. 27. (x = 0, 1, · · · , n)
  28. 28. # p = 0.7 trial_size = 10000 sample_size = 30 set.seed(71) # gen_binom_var <- function() { return(sum(rbern(sample_size, p))) } result <- rdply(trial_size, gen_binom_var()) # dens <- data.frame(y=dbinom(seq(sample_size), sample_size, 0.7))*trial_size # ggplot() + layer(data=resuylt, mapping=aes(x=V1), geom="bar", stat = "bin", binwidth=1, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=seq(sample_size)+.5, y=y), geom="line", stat="identity", position="identity",colour="red" ) + ggtitle("Bernoulli to Binomial.")
  29. 29. P(X = x) = e x x!
  30. 30. trial_size = 5000; width <- 1; # p = 0.7; n = 10; np <- p*n # n!∞ p!0 np= n = 100000; p <- np/n # gen_binom_var <- function() { return(sum(rbern(n, p))) } result <- rdply(trial_size, gen_binom_var()) # dens <- data.frame(y=dpois(seq(20), np))*trial_size # ggplot() + layer(data=result, mapping=aes(x=V1), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=seq(20)+.5, y=y), geom="line", stat="identity", position="identity", colour="red" ) + ggtitle("Bernoulli to Poisson.")
  31. 31. f(x) = 1 p 2⇡ 2 exp ⇢ 1 2 (x µ)2 2 ( 1 < x < 1)
  32. 32. # n <- 10000; p <- 0.7; trial_size = 10000 width=10 # gen_binom_var <- function() { return(sum(rbern(n, p))) } result <- rdply(trial_size, gen_binom_var()) # dens <- data.frame(y=dnorm(seq(6800,7200), mean=n*p, sd=sqrt(n*p*(1-p)))*trial_size*width) # ggplot() + layer(data=result, mapping=aes(x=V1), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=seq(6800,7200), y=y), geom="line", stat="identity", position="identity", colour="red") + ggtitle("Bernoulli to Normal.")
  33. 33. ( 1 < x < 1) f(x) = 1 p 2⇡ exp ⇢ 1 2 x2
  34. 34. # n <- 10000; p <- 0.7 trial_size = 30000 width=0.18 # gen_binom_var <- function() { return(sum(rbern(n, p))) } result <- rdply(trial_size, gen_binom_var()) m <- mean(result$V1); sd <- sd(result$V1); result <- (result - m)/sd # dens <- data.frame(y=dnorm(seq(-4,4,0.05), mean=0, sd=1)*trial_size*width) # ggplot() + layer(data=result, mapping=aes(x=V1), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=seq(-4,4,0.05), y=y), geom="line", stat="identity", position=“identity", colour="red" ) + ggtitle("Bernoulli to Standard Normal.")
  35. 35. f(x, k) = (1/2)k/2 (k/2) xk/2 1 e x/2 (0  x) Xi Z = X2 1 + · · · + X2 k
  36. 36. # p <- 0.7; n <- 1000; trial_size <- 100000; width <- 0.3; df <- 3 # (3 ) gen_binom_var <- function() { return(sum(rbern(n, p))) } gen_chisq_var <- function() { result <- rdply(trial_size, gen_binom_var()) return(((result$V1 - mean(result$V1))/sd(result$V1))**2) } # df result <- rlply(df, gen_chisq_var(),.progress = "text") res <- data.frame(x=result[[1]] + result[[2]] + result[[3]]) # ( =3) xx <- seq(0,20,0.1) dens <- data.frame(y=dchisq(x=xx, df=df)*trial_size*width) # ggplot() + layer(data=data, mapping=aes(x=x), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=xx, y=y), geom="line", stat="identity", position="identity", colour="blue" ) + ggtitle("Bernoulli to Chisquare")
  37. 37. f(x, ) = ⇢ e x (x 0) 0 (x < 0)
  38. 38. trial_size = 7000; width <- .01; # p = 0.7; n = 10; np <- p*n; # n!∞ p!0 np= n = 10000; p <- np/n # gen_exp_var <- function() { cnt <- 0 while (TRUE) { cnt <- cnt + 1 if (rbern(1, p)==1){ return(cnt) # 1 } } } data <- data.frame(x=rdply(trial_size, gen_exp_var())/n) names(data) <- c("n", "x") # dens <- data.frame(y=dexp(seq(0, 1.5, 0.1), np)*trial_size*width) ggplot() + layer(data=data, mapping=aes(x=x), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=seq(0, 1.5, 0.1), y=y), geom="line", stat="identity", position="identity", colour="red" ) + ggtitle("Bernoulli to Exponential.")
  39. 39. f(x, ↵, ) = ↵ (↵) x↵ 1 exp( x) (0  x < 1) ↵X i=1 Xi ⇠ (↵, )Xi ⇠ Exp( )
  40. 40. trial_size = 7000; width <- .035; # p = 0.7; n = 10; np <- p*n; # n!∞ p!0 np= n = 10000; p <- np/n; alpha <- 5 # get_interval <- function(){ cnt <- 0 while (TRUE) { cnt <- cnt + 1 if (rbern(1, p)==1){ return(cnt) } } } gen_exp_var <- function() { data <- data.frame(x=rdply(trial_size, get_interval())/n) names(data) <- c("n", "x") return(data) } result <- rlply(alpha, gen_exp_var()) data <- data.frame(x=result[[1]]$x + result[[2]]$x + result[[3]]$x + result[[4]]$x + result[[5]]$x) # dens <- data.frame(y=dgamma(seq(0, 3,.01), shape=alpha, rate=np)*trial_size*width) ggplot() + layer(data=data, mapping=aes(x=x), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=seq(0,3,.01), y=y), geom="line", stat="identity", position="identity", colour="red" ) + ggtitle("Bernoulli to Gamma")
  41. 41. f(x, ↵, ) = ↵ (↵) x (↵+1) exp ✓ x ◆ (0  x < 1) Xi ⇠ Exp( ) Z = ↵X i=1 Xi ⇠ (↵, ) 1/Z ⇠ IG(↵, )
  42. 42. trial_size = 7000; width <- .; # p = 0.7; n = 10; np <- p*n; # n!∞ p!0 np= n = 10000; p <- np/n; alpha <- 5 # get_interval <- function(){ cnt <- 0 while (TRUE) { cnt <- cnt + 1 if (rbern(1, p)==1){ return(cnt) } } } gen_exp_var <- function() { data <- data.frame(x=rdply(trial_size, get_interval())/n) names(data) <- c("n", "x") return(data) } result <- rlply(alpha, gen_exp_var()) data <- data.frame(x=1/(result[[1]]$x + result[[2]]$x + result[[3]]$x + result[[4]]$x + result[[5]]$x)) # dens <- data.frame(y=dinvgamma(seq(0, 23,.01), shape=5, rate=1/np)*trial_size*width) ggplot() + layer(data=data, mapping=aes(x=x), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=seq(0,3,.01), y=y), geom="line", stat="identity", position="identity", colour="red" ) + ggtitle("Bernoulli to Inversegamma")
  43. 43. f(x) = ⇢ 1 (0  x  1) 0 (otherwise)
  44. 44. Z = x1(1/2)1 + x2(1/2)2 + · · · + xq(1/2)q
  45. 45. width <- 0.02 p <- 0.5; sample_size <- 1000 trial_size <- 100000 gen_unif_rand <- function() { # sample_size 2 # return (sum(rbern(sample_size, p) * (rep(1/2, sample_size) ** seq(sample_size)))) } gen_rand <- function(){ return( rdply(trial_size, gen_unif_rand()) ) } system.time(res <- gen_rand()) ggplot() + layer(data=res, mapping=aes(x=V1), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + ggtitle("Bernoulli to Standard Uniform")
  46. 46. f(x, a, b) = ⇢ (b a) 1 (a  x  b) 0 (otherwise)
  47. 47. a <- 5 b <- 8; width <- 0.05 p <- 0.5 sample_size <- 1000 trial_size <- 500000 gen_unif_rand <- function() { # sample_size 2 # return (sum(rbern(sample_size, p) * (rep(1/2, sample_size) ** seq(sample_size)))) } gen_rand <- function(){ return( rdply(trial_size, gen_unif_rand()) ) } system.time(res <- gen_rand()) res$V1 <- res$V1 * (b-a) + a ggplot() + layer(data=res, mapping=aes(x=V1), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + ggtitle("Bernoulli to Uniform") + xlim(4,9)
  48. 48. f(x, ↵, ) = 1 B(↵, ) x↵ 1 (1 x) 1 (0 < x < 1) Xi ⇠ U(0, 1)iid (i = 1, 2, · · · , ↵ + 1)
  49. 49. width <- 0.03; p <- 0.5 digits_length <- 30; set_size <- 3 trial_size <- 30000 gen_unif_rand <- function() { # digits_length 2 # return (sum(rbern(digits_length, p) * (rep(1/2, digits_length) ** seq(digits_length)))) } gen_rand <- function(){ return( rdply(set_size, gen_unif_rand())$V1 ) } unif_dataset <- rlply(trial_size, gen_rand, .progress='text') p <- ceiling(set_size * 0.5); q <- set_size - p + 1 get_nth_data <- function(a){ return(a[order(a)][p]) } disp_data <- data.frame(lapply(unif_dataset, get_nth_data)) names(disp_data) <- seq(length(disp_data)); disp_data <- data.frame(t(disp_data)) names(disp_data) <- "V1" x_range <- seq(0, 1, 0.001) dens <- data.frame(y=dbeta(x_range, p, q)*trial_size*width) ggplot() + layer(data=disp_data, mapping=aes(x=V1), geom="bar", stat = "bin", binwidth=width, fill="#6666ee", color="gray" ) + layer(data=dens, mapping=aes(x=x_range, y=y), geom="line", stat="identity", position="identity", colour="red" ) + ggtitle("Bernoulli to Beta")
  50. 50. E[X] = X( )P( ) + X( )P( ) = 0 ⇥ 0.8 + 1, 000, 000 ⇥ 0.2 = 200, 000 E[X] = X x xp(x) µ
  51. 51. ✓ n x ◆ = n! (n x)!x! E[X] = nX x=0 xP(x) = nX x=0 x ✓ n x ◆ px (1 p)n x = nX x=0 x n! (n x)!x! px (1 p)n x = nX x=0 n (n 1)! (n x)!(x 1)! px (1 p)n x = np nX x=0 ✓ n 1 m 1 ◆ p(x 1) (1 p)(n 1) (x 1) = np = np nX x=1 ✓ n 1 m 1 ◆ p(x 1) (1 p)(n 1) (x 1) = np
  52. 52. Var[X] = E[(X E[X])2 ] = X x (x E[x])2 P(x) = 2 µ
  53. 53. Var[x] = E[(X E[X])2 ] = Z 1 1 (x E[x])2 f(x)dx = 2 E[X] = Z 1 1 xf(x)dx = µ
  54. 54. E[g(X)] = Z 1 1 g(x)f(x)dx g(X) = (X E[X])2 E[ · ] = Z 1 1 · f(x)dx
  55. 55. g(x) = xk E[g(X)] = E[Xk ] = Z 1 1 xk f(x)dx µ0 k
  56. 56. g(x) = (x E[x])k E[g(X)] = E[(X E[X]])k ] = Z 1 1 (x E[x])k f(x)dx µk
  57. 57. E[cX] = cE[X] * E[cX] = Z 1 1 cxf(x)dx = c Z 1 1 xf(x)dx = cE[X]
  58. 58. Var[cX] = c2 Var[X] * Var[cX] = Z 1 1 (cx E[cx])2 f(x)dx = Z 1 1 (cx cµ)2 f(x)dx = Z 1 1 c2 (x µ)2 f(x)dx = c2 Z 1 1 (x µ)2 f(x)dx = c2 Var[X]
  59. 59. P(x < X 5 x + x, y < Y 5 y + y) x, y ! 0 f(x, y) = lim x, y!0 P(x < X 5 x + x, y < Y 5 y + y) f(x, y)
  60. 60. g(x) = Z 1 1 f(x, y)dy h(y) = Z 1 1 f(x, y)dx g(x) h(y)
  61. 61. EX,Y [ g(X, Y )] = Z 1 1 Z 1 1 g(x, y)f(x, y)dxdy g(x, y) = x0.8 y0.8 (x, y) ⇠ N((4, 4), S) S =  1 0.5 0.4 1 EX,Y [ g(X, Y )] = 8.02
  62. 62. g(X, Y ) = (X µX)(Y µY ) Cov[X, Y ] = E[(X µX)(Y µY )]
  63. 63. g(X, Y ) = (X µX)(Y µY ) µX µX µX µX µY µY µY µY S1 = S2 = S3 = S4 =  1 0.8 0.8 1  1 0.8 0.8 1  1 0 0 1  1 0.999 0.999 1 Cov[X, Y ] = E[(X µX)(Y µY )] (x, y) ⇠ N((4, 4), S)
  64. 64. f(x, y) f(x, y) = g(x)h(y)
  65. 65. f(x, y) = g(x)h(y) = 0
  66. 66. (x1, x2, · · · , xn) x1 f(x1) = Z · · · Z f(x1, · · · , xn)dx2 · · · dxn x1 f(x1, · · · , xn) = f(x1) · · · f(xn) x1 · · · xn
  67. 67. x1 · · · xn g1(x1), · · · , gn(xn) x1 · · · xn E[ nY i=1 gi(xi)] = nY i=1 E[gi(xi)] E[g1(x1)] E[gn(xn)] E[ nY i=1 gi(xi)] = Z 1 1 · · · Z 1 1 g1(x1) · · · gn(xn)f(x1, · · · , xn)dx1 · · · dxn = Z 1 1 g1(x1)f(x1)dx1 · · · Z 1 1 gn(xn)f(xn)dxn = nY i=1 E[gi(xi)] f(x1) · · · f(xn)
  68. 68. x1 · · · xn xi µi 2 i i = 1, 2, · · · , n c = (c1, · · · , cn) c1x1 + · · · + cnxn c1µ1 + · · · + cnµn c2 1 2 1 + · · · + c2 n 2 n
  69. 69. E[c1x1 + · · · + cnxn] = Z 1 1 · · · Z 1 1 (c1x1 + · · · + cnxn)f(x1 · · · , xn)dx1 · · · dxn = c1 Z 1 1 · · · Z 1 1 x1f(x1 · · · , xn)dx1 · · · dxn · · · cn Z 1 1 · · · Z 1 1 xnf(x1 · · · , xn)dx1 · · · dxn =c1 Z 1 1 x1dx1 · · · cn Z 1 1 xndxn =c1µ1 + · · · + cnµn f(x1) · · · f(xn) f(x1) · · · f(xn) µ1 µn =c1 Z 1 1 x1dx1 · · · cn Z 1 1 xndxn =c1µ1 + · · · + cnµn
  70. 70. Var[c1x1 + · · · + cnxn] = E[{(c1x1 + · · · + cnxn) E[c1x1 + · · · + cnxn]}2 ] = E[{c1(x1 µ1) + · · · + c1(x1 µ1)}2 ] = E[ nX i=1 c2 i (xi µi)2 + X i6=j cicj(xi µj)(xi µj)] = nX i=1 c2 i E[(xi µi)2 ] + X i6=j cicjE[(xi µj)(xi µj)] = c2 1 2 1 + · · · + c2 n 2 n c1µ1 + · · · + cnµn = E[xi µi]E[xj µj] = 0= 2 i
  71. 71. x1 · · · xn x1 · · · xn xi µ 2 (µ, 2 )
  72. 72. x1 · · · xn T = x1 + · · · + xn E[T] = E[x1 + · · · + xn] = E[x1] + · · · + E[xn] = nµ Var[T] = Var[x1 + · · · + xn] = Var[x1] + · · · + Var[xn] = n 2 2 1 = · · · = 2 n c1 = · · · = cn = 1 Var[c1x1 + · · · + cnxn] = c2 1 2 1 + · · · + c2 n 2 n
  73. 73. ¯x = 1 n nX i=1 xi = 1 n T E[¯x] = 1 n E[T] = n · 1 n µ = µ Var[¯x] = Var[ 1 n T] = 1 n2 Var[T] = 2 n µ 2
  74. 74. Var[¯x] = 2 n = 0.0833 500 = 0.000166 E[¯x] = 0.5
  75. 75. Var[¯x]
  76. 76. µ 2 P(|x µ| > ) 5 1 2 µ 2 1/ 2 = 1 ) P(|x µ| > ) 5 1 = 2 ) P(|x µ| > ) 5 1/4 = 3 ) P(|x µ| > ) 5 1/9
  77. 77. 2 = Z 1 1 (x µ)2 f(x)dx = Z I1 (x µ)2 f(x)dx + Z I2 (x µ)2 f(x)dx + Z I3 (x µ)2 f(x)dx 2 = Z I1 (x µ)2 f(x)dx + Z I3 (x µ)2 f(x)dx = Z I1 2 2 f(x)dx + Z I3 2 2 f(x)dx = 2 2 [P(x 2 I1) + P(x 2 I3)] I1 = ( 1, µ ), I2 = [µ , µ + ], I3 = (µ + , 1) = P(|x µ| > ) P(|x µ| > ) 5 1 2 )
  78. 78. x1 · · · xn µ 2 " > 0 lim n!1 P{|¯xn µ| = "} = 0 ¯xn = 1 n nX i=1 xi ¯xn µ ¯xn ! µ in P
  79. 79. " > 0 P(|¯xn µ| > ") = P(|¯xn µ| > " p n p n ) 5 2 "2n = 2 ¯x= = 1 2
  80. 80. f(x) = 1 p 2⇡ 2 exp ✓ (x µ)2 2 2 ◆ f(x) = 1 p 2⇡ exp ✓ x2 2 ◆ 1 < x < 1 1 < x < 1
  81. 81. f(y) = y2
  82. 82. f(x) = x2 f(y) = y2
  83. 83. f(y) = exp( y2 )
  84. 84. z = p 2y f(z) = exp ✓ 1 2 z2 ◆
  85. 85. Z 1 1 e y2 dy = p ⇡ Z 1 1 exp ✓ z2 2 ◆ dz = p 2⇡ Z 1 1 1 p 2⇡ exp ✓ z2 2 ◆ dz = 1 dz = p 2dy
  86. 86. Z 1 1 1 p 2⇡ exp ✓ z2 2 ◆ dz
  87. 87. z = x µ dz dx = 1 f(x) = Z 1 1 1 p 2⇡ 2 exp ✓ (x µ)2 2 2 ◆ dx 1/
  88. 88. D = (x1, · · · , xn) µ 2 ¯x µ / p n , n ! 1 N(0, 1) = 0.1, µ = 1 = 10, 2 = 1 2 = 100 ¯x = p n = r 1 2n = r 1 0.01 ⇥ 10000 = r 1 100 = 1 10
  89. 89. g(x) = ext E[ext ] = Z 1 1 ext f(x)dx Mx(t) = E[ext ] Mx(t) My(t) x t = 0 y
  90. 90. g(x) = ext ext = 1 + xt + t2 2! x2 + · · · + tk k! xk + · · · Mx(t) = E[ext ] = E[1 + xt + t2 2! x2 + · · · + tk k! xk + · · · ] = 1 + tE[x] + t2 2! E[x2 ] + · · · + tk k! E[xk ] + · · · = 1 + xµ0 1 + t2 2! µ0 2 + · · · + tk k! µ0 k + · · ·
  91. 91. Mx(t) d dtk Mx(t) = E[xk ext ] t = 0 d dtk Mx(0) = E[xk ] = µ0 k
  92. 92. x ⇠ N(µ, ) Mx(t) = E[ext ] = Z 1 1 ext 1 p 2⇡ 2 exp ✓ 1 2 (x µ)2 2 ◆ dx z = x µ x = µ + z dx = dz
  93. 93. Mx(t) = Z 1 1 e(µ+ z)t 1 p 2⇡ 2 exp ✓ 1 2 z2 ◆ dz = eµt Z 1 1 1 p 2⇡ exp ✓ tz 1 2 z2 ◆ dz = eµt Z 1 1 1 p 2⇡ exp ✓ 1 2 [z2 2 tz 2 t2 + 2 t2 ] ◆ dz = eµt Z 1 1 1 p 2⇡ e 2t2 2 exp ✓ 1 2 (z t)2 ◆ dz = eµt e 2t2 2 Z 1 1 1 p 2⇡ exp ✓ 1 2 (z t)2 ◆ dz z t = w dz = dw Mx(t) = eµt e 2t2 2 Z 1 1 1 p 2⇡ exp ✓ w2 2 ◆ dw = eµt+ 2t2 2
  94. 94. (f · g)0 = f0 · g + f · g0 (f g)0 (x) = f0 (g(x))g0 (x) M0 x(t) = (µ + 2 t)eµt+ 2t2 2 M00 x (t) = (µ + 2 t)2 ⇣ eµt+ 2t2 2 ⌘ + 2 ⇣ eµt+ 2t2 2 ⌘ = ⇣ eµt+ 2t2 2 ⌘ {(µ + 2 t)2 + 2 }
  95. 95. Var[x] = E[x2 ] (E[x])2 = (µ2 + 2 ) (µ)2 = 2 Var[x] = E[(x E[x])2 ] = E[x2 2E[x]x + E[x]2 ) = E[x2 ] 2E[x]2 + E[x]2 = E[x2 ] E[x]2 t = 0 E[x] = M0 x(0) = (µ + 2 · 0)eµ·0+ 2·02 2 = µ E[x2 ] = M00 x (0) = ⇣ eµ·0+ 2·02 2 ⌘ {(µ + 2 · 0)2 + 2 } = µ2 + 2
  96. 96. D = (x1, · · · , xn) µ 2 ¯x µ / p n , n ! 1 N(0, 1) T = x1 + · · · + xn T nµ p n 2T0 = T nµ p n = ¯x µ 1/ p n
  97. 97. Mx(t) My(t) x t = 0 y T T0 = T nµ p n N(0, 2 )
  98. 98. Mxi (t) = 1 + µ0 1t + µ0 2 t2 2! + µ0 3 t3 3! + · · · Mxi µ(t) = 1 + µ1t + µ2 t2 2! + µ3 t3 3! + · · · = 1 + 0 + 2 t2 2! + µ3 t3 3! + · · ·
  99. 99. xi µ p n xi µ p n Mxi µ p n (t) = E[e xi µ p n t ] = 1 + 2 t2 2!n + µ3 t3 3!n3/2 + · · · + µk tk k!nk/2 + · · · = 1 + 2 t2 2n + n 2n = 1 2n n n ! 0 n ! 0 = 1 + 2 t2 + n 2n
  100. 100. T0 = x1 µ p n + x2 nµ p n + · · · + xn µ p n = nX i=1 xi µ p n MT 0 (t) = MPn i=1 ⇣ xi µ p n ⌘(t) = E[e Pn i=1 ⇣ xi µ p n ⌘ t ] = nY i=0 E[e ⇣ xi µ p n ⌘ t ] = ✓ 1 + 1 n 2 t2 + n 2 ◆n er ⌘ lim n!1 ⇣ 1 + r n ⌘n r r = lim n!1 ⇣ 1 + r n ⌘n
  101. 101. n ! 1 lim n!1 MT 0 = lim n!1 ✓ 1 + 1 n 2 t2 + n 2 ◆n = e 2t2 2 lim n!1 n = 0 N(0, 2 ) T0 = T nµ p n 2
  102. 102. n = 100000 sample_size = 1000 rvs_list = [] m_list = [] for i in range(n): unif_rvs = st.uniform.rvs(4.5, size=sample_size) # 5 beta_rvs = st.beta.rvs(a=3, b=3, size=sample_size) # 0.5 β gamma_rvs = st.gamma.rvs(a=3, size=sample_size) # 3 chi2_rvs = st.chi2.rvs(df=5, size=sample_size) # exp_rvs = st.expon.rvs(loc=0, size=sample_size) # 1 rvs = np.array([unif_rvs, beta_rvs, gamma_rvs, chi2_rvs, exp_rvs]).flatten() m_list.append(np.mean(rvs)) rvs_list.append(rvs)
  103. 103. # n = 10000 sample_size = 1000 rvs_list = [] m_list = [] m_unif = st.uniform.rvs(4, 2, size=sample_size) m_beta_a = st.uniform.rvs(4, 2, size=sample_size) m_beta_b = st.uniform.rvs(4, 2, size=sample_size) m_gamma = rd.randint(2,5,size=sample_size) m_chi2_df = rd.randint(3,6,size=sample_size) m_exp = st.uniform.rvs(4, 2, size=sample_size) def gen_random_state(): return int(dt.now().timestamp() * 10**6) - 1492914610000000 + rd.randint(0, 1000000) def create_rvs(n): #rd.seed = int(dt.now().timestamp() * 10**6) - 1492914610000000 + rd.randint(0, 1000000) print("[START]") for _ in range(n): unif_rvs = [st.uniform.rvs(m, size=1, random_state=gen_random_state()) for m in m_unif] # 5 beta_rvs = [st.beta.rvs(a=a, b=b, size=1, random_state=gen_random_state()) for a, b in zip(m_beta_a, m_beta_b)]# 0.5 β gamma_rvs = [st.gamma.rvs(a=a, size=1, random_state=gen_random_state()) for a in m_gamma] # 3 chi2_rvs = [st.chi2.rvs(df=d, size=1, random_state=gen_random_state()) for d in m_chi2_df] # exp_rvs = [st.expon.rvs(loc=l, size=1, random_state=gen_random_state()) for l in m_exp] # 1 rvs = np.array([unif_rvs, beta_rvs, gamma_rvs, chi2_rvs, exp_rvs]).flatten() l_mean.append(np.mean(rvs)) l_rvs.append(rvs) print("[END]")
  104. 104. n_jobs = 20 n_each = int(n/n_jobs) jobs = [Process(target=create_rvs, args=(n_each,)) for _ in range(n_jobs)] manager = Manager() l_rvs = manager.list(range(len(jobs))) l_mean = manager.list(range(len(jobs))) start_time = time.time() for j in jobs: j.start() time.sleep(0.2) for j in jobs: j.join() finish_time = time.time() print(finish_time - start_time) m_list = l_mean[n_jobs:] rvs_list = np.array(l_rvs[n_jobs:]) print(rvs_list.shape)
  105. 105. D = (x1, · · · , xn)
  106. 106. ✓0 = ˆ✓(X1, · · · , Xn) ˆ✓lower(X1, · · · , Xn) 5 ✓0 5 ˆ✓upper(X1, · · · , Xn)
  107. 107. ˆ✓(X)
  108. 108. E[(ˆ✓(X) ✓)2 ]
  109. 109. E[(ˆ✓(X) ✓)2 ] = E[{(E[ˆ✓(X)] ✓) + (ˆ✓(X) E[ˆ✓(X)])}2 ] = E[(E[ˆ✓(X)] ✓)2 + 2(E[ˆ✓(X)] ✓)(ˆ✓(X) E[ˆ✓(X)]) + (ˆ✓(X) E[ˆ✓(X)])2 ] = (E[ˆ✓(X)] ✓)2 + Var[ˆ✓(X)] E[ˆ✓(X)] ✓ E[(ˆ✓(X) ✓)2 ] = Var[ˆ✓(X)]
  110. 110. E[¯x] = 1 n E[T] = n · 1 n µ = µ ¯x s2 = 1 n 1 nX i=1 (xi ¯x)2
  111. 111. lim n!1 P{|¯xn µ| = "} = 0 ¯xn ! µ in P ˆ✓n(X) n ! 1 ˆ✓n(X) ! ✓ in P ˆ✓n(X) ¯xn µ
  112. 112. Var[ˆ✓(X)] ˆ✓(X)
  113. 113. D = (x1, · · · , xn) xi f(xi) nY i=1 f(xi) nY i=1 f(xi|✓) xi `(✓|x1, x2, · · · , xn) = nY i=1 f(xi|✓)
  114. 114. x1, x2, · · · , x10 f(x1, x2, · · · , x10|µ, 2 ) = 10Y i=1 1 p 2⇡ 2 exp ✓ 1 2 (xi µ)2 2 ◆
  115. 115. `(µ, 2 |x1, x2, · · · , x10) = 10Y i=1 1 p 2⇡ 2 exp ✓ 1 2 (xi µ)2 2 ◆
  116. 116. ✓⇤ = arg max ✓ `(✓|x1, x2, · · · , xn) log `(✓|x1, · · · , xn) ⌘ L(✓|x1, · · · , xn) `
  117. 117. µ, 2 L(µ, 2 |x1, x2, · · · , x10) = n 2 (2⇡) n 2 log 2 1 2 2 nX i=1 (xi µ)2 @L @µ = 1 2 2 nX i=1 (xi µ)2 ) nX i=1 xi = nµ ) µ⇤ = 1 n nX i=1 xi `(µ, 2 |x1, x2, · · · , xn) = nY i=1 1 p 2⇡ 2 exp ✓ 1 2 (xi µ)2 2 ◆
  118. 118. @L @ 2 = n 2 1 2 + 1 2( 2)2 nX i=1 (xi µ)2 = 0 ) 1 2( 2)2 nX i=1 (xi µ)2 = n 2 2 ) 2⇤ = 1 n nX i=1 (xi µ)2 2⇤
  119. 119. D = (x1, · · · , xn)µ 2 µ
  120. 120. u ⇠ N(0, 1) t = u p v/m v ⇠ 2 (m) f(t) = m+1 2 p m⇡ m 2 ✓ t2 m + 1 ◆ m+1 2
  121. 121. u ⇠ N(0, 1) v ⇠ 2 (m) v > 01 < u < +1 f(u, v) = 1 p 2⇡ exp ✓ u2 2 ◆ (1/2)n/2 (n/2) vn/2 1 e v/2 t = u p v/m x = v f(t) = m+1 2 p m⇡ m 2 ✓ t2 m + 1 ◆ m+1 2 (z) = Z 1 0 tz 1 e t dt
  122. 122. µ D = (x1, · · · , xn) xi ⇠ N(µ, 2 ) ¯x ⇠ N(µ, 2 /n)¯x 1 2 nX i=1 (xi ¯x)2 ⇠ 2 n 1
  123. 123. u = ¯x µ / p n ⇠ N(0, 1) v = 1 2 nX i=1 (xi ¯x)2 ⇠ 2 n 1 t = u p v/(n 1) = ¯x µ / p n · " 1 2 1 (n 1) nX i=1 (xi ¯x)2 # 1/2 = ¯x µ 1/ p n · 1 p s2 = ¯x µ s/ p n ⇠ tn 1 s2 = 1 n 1 nX i=1 (xi ¯x)2 s2
  124. 124. P ✓ tn 1;↵/2 5 ¯x µ s/ p n 5 tn 1;↵/2 ◆ = 1 ↵ tn 1;↵/2 tn 1;↵/2 ↵/2 ↵/2 1 ↵ 1 ↵ 1 ↵ P ✓ ¯x tn 1;↵/2 s p n 5 µ 5 ¯x + tn 1;↵/2 s p n ◆ = 1 ↵ [ tn 1;↵/2, tn 1;↵/2] µ 1 ↵
  125. 125. P ✓ tn 1;↵/2 5 ¯x µ s/ p n 5 tn 1;↵/2 ◆ = 1 ↵ tn 1;↵/2 tn 1;↵/2 ↵/2 ↵/2 1 ↵ 1 ↵ 1 ↵ P ✓ ¯x tn 1;↵/2 s p n 5 µ 5 ¯x + tn 1;↵/2 s p n ◆ = 1 ↵ [ tn 1;↵/2, tn 1;↵/2] µ 1 ↵
  126. 126. = 1 µ = 0 H0 : µ0 = 0 H1 : µ 6= µ0
  127. 127. ¯x = / p n / p 10 ; /3.16
  128. 128. ↵/2 ↵/2 H0 : µ0 = 0
  129. 129. H1 : µ = 1
  130. 130. H1 : µ = 0.5
  131. 131. H1 : µ = 3 µ0H1 : µ = 3 H0 : µ0 = 0
  132. 132. e↵ect size : = µ µ0
  133. 133. … … … … … … … …
  134. 134. r = 1 n Pn i=1(xi ¯x)(yi ¯y) q 1 n Pn i=1(xi ¯x)2 q 1 n Pn i=1(yi ¯y)2
  135. 135. r = 1 n Pn i=1(xi ¯x)(yi ¯y) q 1 n Pn i=1(xi ¯x)2 q 1 n Pn i=1(yi ¯y)2
  136. 136. r = 1 n Pn i=1(xi ¯x)(yi ¯y) q 1 n Pn i=1(xi ¯x)2 q 1 n Pn i=1(yi ¯y)2
  137. 137. r = 1 n Pn i=1(xi ¯x)(yi ¯y) q 1 n Pn i=1(xi ¯x)2 q 1 n Pn i=1(yi ¯y)2
  138. 138. 1 n nX i=1 (xi ¯x)(yi ¯y)
  139. 139. 1 n nX i=1 (xi ¯x)(yi ¯y)
  140. 140. 1 n nX i=1 (xi ¯x)(yi ¯y)
  141. 141. 1 n nX i=1 (xi ¯x)(yi ¯y)
  142. 142. 1 n nX i=1 (xi ¯x)(yi ¯y)
  143. 143. 1 n nX i=1 (xi ¯x)(yi ¯y)
  144. 144. 1 n nX i=1 (xi ¯x)(yi ¯y)

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