スチューデントのt分布 フォンミーゼス分布 混合ガウス分布

2. 97
.
2 3
8

4. !"
#(%|', ), *) = -
.
/
0(%|', 123)4)5(1|), *)61
= 7
.
/
*89 2:; 1823
)<()
1
2>
3
?
exp{−
1
2
% − ' ?
}61
=
<() +
1
2
)*8
)<()
1
2>
3
?
* +
% − ' ?
2
282
3
?
'
,
H = 2) I = a/b ,

5. !
" !
#
$
%!('|$, ", #) =
,-(
#
2
+
1
2
)-( ⁄# 2
"
2#
⁄3 4
1 +
" ' − $ 4
#
6 ⁄7 46 ⁄3 4

6. !
!
2
! 4 7 )#(%|', )*+
1(. )
,-(%|', ), !) =
/0(
!
2
+
1
2
)0( ⁄! 2
)
5!
⁄+ 6
1 +
) % − ' 6
!
* ⁄8 6* ⁄+ 6

8. !
"#$ =
1
'
(
)*+
,
-)
!
( )
.

9. ! 2
. 5
"
#$%
&
)((*#|,(-#, /), 01%
1
"
#$%
&
)2*(*#|,(-#, /), 3, 4
5 2
* 1
1

10. !
"!($|&, (, )) = ,
-
.
/($|&, 0( 12
)345(0| ⁄) 2 , ⁄) 2)80
"!(9|:, ;, )) = ,
-
.
/(9|:, 0; 12
)345(0| ⁄) 2 , ⁄) 2)80
"!(9|:, ;, )) =
<=(
)
2
+
?
2
)=( ⁄) 2
|;| ⁄2 @
A) ⁄2 @
1 +
C@
)
1 ⁄D @1 ⁄E @
)C@
= 9 − : G
;(9 − :
H($|&, 4, I) = ,
-
.
/($|&, J12)345(J|4, I)8J
) = 24 ( = a/b 0 = JI/4
,

11. ![#] = & '
cov # =
+
+,-
.,/
'
0123[#] = &
>
2 >1

12. !", $

13. 4
1 6
2

14. •
!" = 1∘
, !' = 359∘
0∘
1180∘
179∘
0∘
10∘
1∘
0 8
!" = 1∘ !' = 359∘

15. 1{ , , }ND q q= !
!" =
1
%
&
'()
*
"'
)"' = (cos0', sin0'
)!" = ( ̅5cos ̅0, ̅5sin ̅0
̅6) = ̅5cos ̅0 =
1
%
&
'()
*
cos0' ̅67 = ̅5sin ̅0 =
1
%
&
'()
*
sin0'
tan0 =
:;<=
>?:=
̅0
:
,.‖"'‖ = 1
̅0 = tanA)
∑' sin0'
∑' cos0'
C0) + ⋯ 0' ⁄) %

16. !(#) 2
!(#) ≥ 0
'
(
)*
)!(# +# = 1
)!(# + 201) = !(#
!(23, 2)) =
1
215)
exp −
23 − :3
) + 2) − :)
)
25)
; = :3, :) , < = 5)=
)> = (23, 2)
!(>)

17. !(#$, #&) =
1
2+,&
exp −
#$ − 1$
& + #& − 1&
&
2,&
1$ 34cos84 1& 34sin84
#$ rcos8 #& rsin8
=1
, 3
−
1
2,&
3cos8 − 34cos84
& + 3sin8 − 34sin84
&
= −
1
2,&
1 + 34
& − 234cos8cos84 − 234sin8sin84
=
34
,& cos(8 − 84) + <=>?@.
cos&
A + sin&
A = 1
)cosCcosD + sinCsinD = cos(C − D
!(8|84, F) =
1
)2+G4(F
exp{Fcos 8 − 84 }
F =
JK
LM !(8)

18. !(#|#%, ') =
1
)2,-%('
exp{'cos # − #% }
-%(') =
1
2,
7
%
89
}exp{'cos# :#
#%
' 5
' -%(')
0 .
2
,
4
3,
4

19. !"
!"
ln%('|!", *) = −.ln(20) − .ln1"(*) + * 3
456
7
)cos(!4 − !"
3
456
7
sin(!4 − !") = 0
sin(= − >) = sin=cos> − cos=sin>
tan! =
sin!
cos!
!"
AB
= tanC6
∑4 sin!4
∑4 cos!4
!"
̅! = tanC6
∑4 sin!4
∑4 cos!4

20. !
• !
)#$
%(!) = #((!
)(!*+) =
1
-
.
/0(
1
)cos(5/ − 5%
*+
)(!) =
)#((!
)#%(!
.
) !*+ =
1
-
.
/0(
1
cos5/ cos5%
*+
+
1
-
.
/0(
1
sin5/ sin5%
*+
= ̅< ( 2.55)
ln@(A|5%, !) = −-ln(2D) − -ln#%(!) + ! .
/0(
1
)cos(5/ − 5%
!
#%(!) =
1
2D
E
%
FG
}exp{!cos5 M5
r

21. !" # ,
!"
$%
= tan*+
∑- sin!-
∑- cos!-
2 #$% = ̅4
,

22. .
• 5 2
• . ,
•
0 2#

24. ,

25. !(#) = &
'()
*
)+',(#|.', 0'
1
+': :
),(#|.', 0' :

26. !" =
!$ =
= !%
&(()
&(()

27. !
"#$
%
&" = 1
0 ≤ &" ≤ 1
)&" = ,(.
∫ ,(0)10 = !
"
&" ∫ )2(0|4", 6" 10
)2(0|4", 6" ≥ 0 , ,(0) ≥ 0,
=
&", &", 1 0 . ,

28. )"#(%) ≡ '((|% =
)'(()'(%|(
+,
)'(-)'(%|-
=
).#/(%|0#, 2#
+,
).,/(%|0,, 2,
( %
'(%) = 3
#45
6
)'(()'(%|(
).# = '((
)/(%|0#, 2# = )'(%|(
: %
'(( = 1|% ) .5
)"#(%) ≡ '((|%

29. }" ≡ {%&, ⋯ , %)}, * ≡ {*&, ⋯ , *)}, + ≡ {+&, ⋯ , +)
1X { , , }N= !
1 1
ln (X , ) ln ( | , )
N K
k n k k
n k
p Np µ p µ
= =
ì ü
S = Sí ý
î þ
å å
)-(-/0
1
2 +2

30. ,

31. )"#(%|', ), * * → ∞ )-(%|', )./
%
∝ 1 +
) % − ' 4
*
.
56/
4
)"#(%|', ), *
789: = %= exp ln 1 +
) % − ' 4
*
.
56/
4
= exp −
* + 1
2
ln 1 +
) % − ' 4
*
)ln(1 + B) = B + C(B4 ln 1 + # = D
1
1 + #
E# = F(1 − # + #4 ⋯ )E# = # −
#4
2
+

32. exp −
% + 1
2
) * − + ,
%
+ -(%/,)
= exp −
) * − + ,
2
−
) * − + ,
2%
+ -(%/2)
lim
6→8
exp −
) * − + ,
2
−
) * − + ,
2%
+ -(%/2) = exp(−
) * − + ,
2
) )9(*|+, )/2
)ln(1 + =) = = + -(=,

33. ! "#
$ " "#, ! ! → ∞ ( )
$("|"#, !) =
1
)2./#(!
exp{!cos " − "# }
$(9) ∝ exp ;!cos(! ⁄=> ?9
9 = ! ⁄> ? " − "# )
cos@ = 1 −
@?
2!
+ C(@D);cos(! ⁄=> ?9 = 1 −
!=>9?
2!
+ C(!=?)
@

34. !cos(& ⁄() *+ = 1 −
/0123
*!
+ 6(&(*)
8(+) ∝ exp !&cos(& ⁄() *+
8(+) ∝ exp(m)exp(
−+*
2
+ 6(&()))
+ = & ⁄) * ? − ?@
exp(m) + & → ∞
8(+) ∝ exp(
−+*
2
)
( )p q µ
8(?) ∝ exp(
−& ? − ?@
*
2
) )C(?|?@, &()

35. . !"# $ !"# = ̅'
$ !"# =
1
)
*
+,-
.
cos2+ cos23
"#
+
1
)
*
+,-
.
sin2+ sin23
"#
̅'cos ̅2 =
1
)
*
+,-
.
cos2+ , ̅'sin ̅2 =
1
)
*
+,-
.
sin2+
̅2 23
"#
$ !"# = ̅'cos ̅2cos23
"#
+ ̅'sin ̅2sin23
"#
$ !"# = ̅' cos823
"#
+ sin823
"#
= ̅' ∴ $ !"# = ̅'