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?_?@?̈Ⴂ?ɂ?蓜?x?̂΂???͈قȂ?̂??H

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?ƂȂ?܂??B?s?Ε??U?̔䂪???̂܂?F?l?ɂȂ?Ƃ????킯?ł??B

?@s₁²????s₂²?̒l???傫?????F?l?????????Ȃ?A?ݐϊm???????????Ȃ?܂??B?t?ɁAs₁²????s₂²?̒l???????????F?l???傫???Ȃ?A?E???m???????????Ȃ?܂??B?O?҂͊m?????x?֐??̃O???t?ŁA?????̒l?????????Ȃ????ꍇ?ɓ?????A??҂͑傫???Ȃ????ꍇ?ɓ?????܂??i?}5?j?B

?}5?@F?l?̑召?Ɨݐϊm???A?E???m??
?s?Ε??U?̔?iF?l?j?????????Ȃ?ƁA?ݐϊm???????????Ȃ?B?t?ɕs?Ε??U?̔䂪?傫???Ȃ?ƁA?E???m???????????Ȃ?B?s?Ε??U?̔䂪1?ɋ߂??Ɨݐϊm???̓O???t?̔????̖ʐς?0.5?ɋ߂??Ȃ?B

?@?ł́A?ȉ??̕\1?Ɏ????T???v???Ŋm?F???Ă݂܂??傤?B?ꕪ?U?????????Ɖ??肷??ƁA?s?Ε??U?̔?A?܂?F?l???ɒ[?ɏ??????Ȃ?i?ݐϊm???????????Ȃ?j???Ƃ͂Ȃ??͂??ł??B?t?ɁAF?l???ɒ[?ɑ傫???Ȃ?i?E???m???????????Ȃ?j???Ƃ??Ȃ??͂??ł??B?c?c???A???????????m???̏????Ȃ??Ƃ??N???????̂ł???΁A?ꕪ?U?????????Ƃ???????????i???j?ĂȂ???΂Ȃ?܂???ˁB


?_?@	?T???v??
A	19.3?@20.6?@20.7?@19.7?@19.4?@20.2?@20.3?@20.2?@20.7?@19.5
B	20.1?@19.9?@19.7?@20.4?@20.2?@20.7?@20.4?@19.9?@20.9?@20.4
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?}6?@F?l?̑召?Ɨݐϊm???A?E???m??
?_?@A?Ɣ_?@B?̃T???v???̒l?͏c?????ɓ??͂???Ă???B?s?Ε??U?????߂?Ɣ_?@B?̂΂???????????悤?Ɍ?????B?Z??F4?ŋ??߂??s?Ε??U?̔䂪F?l?ƂȂ?BF?l????ɃZ??F6?ŉE???m???????߂??0.141?i??14.1???j?ƂȂ?B???R?x??9?ł??邱?Ƃɒ??ӁB

?@???ʂ?14.1???ƂȂ?܂??B??ʂɁA5?????邢??1???ȉ??ł???΁u?܂?ɂ????N????Ȃ????Ɓv?ƍl?????A?ꕪ?U?????????Ɖ??肷??ɂ͖?????????i?_?@B?̕ꕪ?U?????????j?Ƃ??????ƂɂȂ?܂??B???????A???̏ꍇ?A??r?I?????Ȓl?ł???Ƃ͂????A?ꕪ?U?????????Ƃ?????????̂Ă?قǂł͂Ȃ??悤?ł??B?܂?A?_?@B?̕ꕪ?U?????????Ƃ͌????Ȃ??Ƃ??????ƂɂȂ?܂??i?Ƃ????Ă??A?????????i???j?Ă??Ȃ??Ƃ????????Ȃ̂ŁA???????ƒf?肷?邱?Ƃ͂ł??܂???j?B

?@???́A???̌v?Z?͓????U?̌???̂??߂̌v?Z?ɂق??Ȃ?܂???B???????A?A????????Η??????Ȃǂ̍l???????\???ɒm??????Ŏg???K?v??????̂ŁA???̎??_?Łu???_?͂??????I?v?ƌ????؂?̂͏????҂??Ă????????B?????U?̌???ɂ??Ă͐??????v?҂ł??b?????܂??B

F???z?̗ݐϕ??z?֐??ɑ΂???t?֐??́H

?@F.INV?֐???F.INV.RT?֐????g???΁AF???z?̗ݐϕ??z?֐??ɑ΂???t?֐??̒l?????߂??܂??B

?@F.INV?֐??ɂ͍????m???Ǝ??R?x1?A???R?x2???w?肵?AF.INV.RT?֐??ɂ͉E???m???Ǝ??R?x1?A???R?x2???w?肵?܂??B?Ⴆ?΁A???R?x1??6?A???R?x2??10??F???z?ŁA?ݐϊm???i?????m???j??95???̂Ƃ???F?l??=F.INV(95%, 6, 10)?ŋ??߂??܂??B???ʂ?3.217?ƂȂ?܂??B???̊m???́A?E???m????5???̂Ƃ???F?l?Ɠ??????Ȃ?̂ŁA=F.INV.RT(5%, 6, 10)?̌??ʂƈ?v???܂??B?󂢂Ă???Z???ɂ????̎?????͂??Ă݂Ċm?F???Ă????Ă????????B

?@?Ȃ??A?????̗?́mF???z?̋t?֐??n???[?N?V?[?g?ɓ??͂???Ă??܂??BF???z?̗ݐϕ??z?֐??ɑ΂???t?֐??̒l?́A?ꕪ?U?̔?̋?Ԑ???ȂǂɎg???܂??B?ڍׂ͐??????v?҂ł??b?????܂??B

?@?????܂ŁAF???z?̊m???ϐ??Ɗm?????x?֐??A?ݐϕ??z?֐??ɂ??Č??Ă??܂????B?܂??A???p??Ƃ??āA?????U?̌???ɂȂ??邨?b?ɂ??i???Ȃ?j???ݍ??݂܂????B????????́A??????V?~?????[?V?????ɂ????F???z?ɂ??Ă̗?????[?߂Ă????????Ǝv???܂??B???p?I?ɂ͒m??Ȃ??Ă????܂??肪?Ȃ??̂ŁA?????????ȕ??́A?Ō?́u???̋L???Ŏ??グ???֐??̌`???v???m?F???āA????͂??J???Ƃ??Ă?????Ă??\???܂???B

?R?????@F???z?ƃJ?C??敪?z?A?s?Ε??U?̊֌W

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# F???z?̊m?????x?֐??̃V?~?????[?V????
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import f

# ???K???zN1?i??=3?A??^2=3?j???烉???_????10?̃T???v???????o???????̂?10000???
x1_varp = 3
x1 = np.random.normal(loc=3, scale=np.sqrt(x1_varp), size=(10000, 10))
# ???K???zN2?i??=4?A??^2=5?j???烉???_????10?̃T???v???????o???????̂?10000???
x2_varp = 5
x2 = np.random.normal(loc=4, scale=np.sqrt(x2_varp), size=(10000, 10))

# ?T???v???̕s?Ε??U???v?Z
x1_vars = np.var(x1, axis=1, ddof=1)
x2_vars = np.var(x2, axis=1, ddof=1)

# F?l?????߂?
F = (x1_vars/x1_varp) / (x2_vars/x2_varp)
# ?q?X?g?O??????`???iF?l???ǂ̂悤?ɕ\??邩????????j
plt.hist(F, bins=100, range=(0, 6), density=True) # ?K????100?A?c???͊m???Ƃ???

# F???z?̊m?????x?֐???`??
x = np.linspace(0, 6, 100) # 0?`6?܂ł?100?ɕ?????????????i?????̒l?j
plt.plot(x, f.pdf(x, 9, 9)) # x?ɑ΂???F???z?̊m?????x?֐??̒l???v???b?g????
plt.show()

???X?g1?@F???z?̊m?????x?֐????V?~?????[?V????????
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?@???X?g1?̎??s?Ⴊ?ȉ??̐}7?ł??B???????g???̂Ō??ʂ͖??񏭂????قȂ?܂????A?قƂ?Ǐd?Ȃ??Ă??܂??ˁB

?}7?@F???z?̊m?????x?֐??̃V?~?????[?V????????
?_?O???t?́A2?̐??K???z????10???̃T???v????10000?????o???A?????̕s?Ε??U?ƕꕪ?U????ɋ??߂?F?l???q?X?g?O?????ɂ??????́B?܂???O???t?͎??R?x1??9?A???R?x2??9??F???z?̊m?????x?֐??B

?R?????@F???z?̊m?????x?֐??Ɨݐϕ??z?֐??𐔎??ŕ\??

?@?ŏ??ɁAF???z?̊m?????x?֐?f(x;k₁,k₂)?Ɨݐϕ??z?֐?F(x;k₁,k₂)??\???????f?ڂ??܂????i?m???ϐ?x?Ǝ??R?xk₁,k₂??;?ŋ?؂??ĕ\?L???Ă??܂??j?B?????????ڂ????m?肽?????̂??߂ɁA??`?ʂ?Ɍv?Z??????@???Љ?Ă????܂??B

?@???????Ak₁,k₂?͎??R?x?AI_z?͐????????ꂽ?s???S?x?[?^?֐??ŁA

?Ƃ??܂??B

?@???͂??̘A?ڂŉ??x???o?ꂵ???K???}?֐??ł??B?K???}?֐??ƕs???S?x?[?^?֐??̒?`?ɂ??ẮA?O??Љ???̂ŁA????????Q?Ƃ??Ă??????????ƂƂ??āA?????ł́AExcel?̊֐????g?????v?Z???@???????????Ă????܂??B?K???}?֐??̒l??GAMMA?֐??ŋ??߂??A?s???S?x?[?^?֐??̒l??BETA.DIST?֐??̗ݐϕ??z?֐??ŋ??߂??܂??B

?@?Ⴆ?΁A???R?x1??6?ŁA???R?x2??10?̏ꍇ?AF??2?ɑ΂???m?????x?֐??̒l???i5?j???̒?`?ɏ]???ċ??߂????ꍇ?́Ak₁??6?Ak₂??10?Ax??2???????邾???ł??B???q??

?@?@=GAMMA((6+10)/2)*(6/10)^(6/2)*2^(6/2-1)

?ŋ??߁A?????

?@?@=GAMMA(6/2)*GAMMA(10/2)*(1+6/10*2)^((6+10)/2)

?ŋ??߂Ċ???Z????΂????ł??ˁB???q??4354.56?A?????26340.42?ƂȂ?̂ŁA???ʂ?0.1653?ƂȂ?܂??B????A?ݐϕ??z?֐??̒l???i6?j???ɏ]???ċ??߂????ꍇ?́ABETA.DIST?֐???z?̒l??a??k₁/2, b??k₂/2?̒l???w?肵?܂??B?]???āA

?@?@=BETA.DIST(6*2/(6*2+10), 6/2, 10/2, TRUE)

?Ɠ??͂??܂??B???ʂ?0.8411?ƂȂ?܂??B?T???v???t?@?C???́mF???z?i??`?ʂ?Ɂj?n???[?N?V?[?g?ɂ͏?̂悤?Ȓl?ł͂Ȃ??A?Z???A?h???X???w?肵??0.0?`6.0?܂ł̊m?????x?֐??̒l?Ɨݐϕ??z?֐??̒l?????߂?????܂߂Ă???܂??B?????Ƃ??A???p?I?ɂ?F.DIST?֐????g?????????֗??ł??ˁB

?@?????F???z?̊m???ϐ??ł???F?l?̈Ӗ??₻?̋??ߕ??A????ɁAF???z?̊m?????x?֐??Ɨݐϕ??z?֐??̋??ߕ??Ȃǂɂ??Ă??b?????܂????B???????v?҂̓??e?ɂ??Ă??A???Ȃ蓥?ݍ???ł??b?????Ă??܂??܂????BF???z?́A?ꕪ?U?̔?̋?Ԑ???ⓙ???U?̌???ȂǂɎg???܂????A?????ł́AF???z?̊m???ϐ??Ɗm?????x?֐??A?ݐϕ??z?֐??ɂ??Ă̗?????[?߂Ă?????????Ώ\?????Ǝv???܂??B

?@????͈?莞?ԓ??ɉ??炩?̎??ۂ??N????m???????߂?ꍇ?ȂǂɎg????w?????z?ɂ??Ă??b?????܂??B????????y???݂ɁI

???̋L???Ŏ??グ???֐??̌`??

?@?֐??̗??p??ɂ??ẮA???̋L???̒??ŏЉ?Ă???ʂ?ł??B?????ł́A??????グ???֐??̊?{?I?ȋ@?\?ƈ????̎w????@???????????Ă????܂??B

F???z?̊m?????x?֐???ݐϕ??z?֐??̒l?????߂邽?߂̊֐?

F.DIST?֐??FF???z?̊m?????x?֐???ݐϕ??z?֐??̒l?????߂?

?`??

F.DIST(x, ???R?x1, ???R?x2, ?֐??`??)

????

x?F ?m???ϐ??̒l?iF?l?j???w?肷??B
???R?x1?F ???R?x1???w?肷??B
???R?x2?F ???R?x2???w?肷??B
?֐??`???F ?ȉ??̒l???w?肷??B
- FALSE ?c?c ?m?????x?֐??̒l?????߂?
- TRUE ?c?c ?ݐϕ??z?֐??̒l?????߂?