Á¦4Àå ÁÖ¿ä³»¿ë°ú Áú¹®
Á¦2Àå°ú
Á¦3Àå¿¡¼´Â ÇÑ °³ÀÇ È®·üº¯¼ö¿¡ ´ëÇÑ ±â´ë°ª, º¯¼öº¯È¯, ºÐÆ÷ µîÀ» °øºÎÇÏ¿´´Âµ¥,
Á¦4Àå¿¡¼´Â ¿©·¯ °³ÀÇ È®·üº¯¼ö, Áï È®·üº¤ÅÍ(random vector)¿¡ ´ëÇØ °øºÎÇÑ´Ù.
4.1 Joint and Marginal Distributions
- univariate models and multivariate models
- n-dimensional random vector
- discrete case: joint pmf, E(g(X,Y)), marginal pmf
- joint distributionÀº marginal distributionÀ» °áÁ¤ÇÏÁö¸¸
±× ¿ªÀº ¼º¸³ÇÏÁö ¾Ê´Â´Ù. (Example 4.1.9)
- continuous case: joint pdf, E(g(X,Y)), marginal pdf.
- Example 4.1.11, 4.1.12: P((X,Y)¡ôA)ÀÇ °è»ê
- F(x,y)¿Í f(x,y)ÀÇ ´ëÀÀ
4.2 Conditional Distributions and Independence
- discrete case: conditional pmf
- The conditional pmf is a pmf, i.e. first, f(y|x)¡Ã0 for
every y, and second, ¢²y
f(y|x) = 1.
- continuous case: conditional pdf
- The conditional pdf is a pdf. [Q] ÀÌ »ç½ÇÀ» ¼ö¸®ÀûÀ¸·Î
Ç¥ÇöÇغ¸ÀÚ.
- E(g(Y)|x)
- E(Y|X) provides the best guess at Y based on knowledge
of X. [Q] ÀÌ »ç½ÇÀ» ¼ö¸®ÀûÀ¸·Î Ç¥ÇöÇغ¸ÀÚ. (Âü°í: ¿¬½À¹®Á¦ 4.13)
- Example 4.2.4: conditional variance; Var(Y|x)=E(Y2|x) -[E(Y|x)]2.
- Example 4.2.4 (continued): X¿Í YÀÇ joint pdf°¡ f(x,y)=exp(-y),
0<x<y<¡Ä·Î Ç¥ÇöµÉ ¼ö ÀÖ´Â ½ÇÁ¦»óȲ (p. 151)
- Y|x ¿Í Y|X ÀÇ ±¸º°, ±×¸®°í E(g(Y)|x) ¿Í E(g(Y)|X) ÀÇ
±¸º°
- definition of 'independence': f(x,y)=f(x)f(y) for every x and y.
- µ¶¸³¼ºÀ» X¿Í YÀÇ marginal pdf(¶Ç´Â pmf)¸¦ ¾ËÁö ¾Ê°íµµ
È®ÀÎÇÒ ¼ö ÀÖ´Â ¹æ¹ý: Lemma 4.2.7°ú ±× Áõ¸í (ÁÖÀÇ: f(x,y)>0ÀÎ ¿µ¿ªÀÌ f(x)>0ÀÎ
¿µ¿ª°ú f(y)>0ÀÎ ¿µ¿ªÀÇ ±³Â÷°ö(cross-product)À¸·Î Ç¥ÇöµÉ ¼ö ¾øÀ¸¸é X¿Í
Y´Â µ¶¸³ÀÌ ¾Æ´Ï´Ù. ¿¹: Example 4.2.4, 4.2.2)
- Theorem 4.2.10: [Q] P(X¡ôA, Y¡ôB)¸¦ E(g(X,Y))·Î Ç¥ÇöÇÏ·Á¸é
g(X,Y)¸¦ ¾î¶»°Ô Á¤ÀÇÇØ¾ß Çϴ°¡? (Áï, Á¤¸® 4.2.10ÀÇ a´Â bÀÇ Æ¯¼öÇÑ °æ¿ì·Î
º¼ ¼ö ÀÖ´Ù.)
- Theorem 4.2.12´Â µ¶¸³ÀÎ µÎ È®·üº¯¼öÀÇ ÇÕÀÇ ºÐÆ÷¸¦ ¾Ë°í
½ÍÀ» ¶§ À¯¿ëÇÏ°Ô ¾²ÀδÙ. (¿¹:Example 4.2.13. [Q] X¢¦Poisson(¥ë), Y¢¦Poisson(¥ì),
X¿Í Y°¡ µ¶¸³À̸é X+YÀÇ ºÐÆ÷´Â Poisson(¥ë+¥ì)ÀÓÀ» Theorem 4.2.12¸¦ ÀÌ¿ëÇÏ¿©
Áõ¸íÇغ¸ÀÚ. Âü°í: pmf¸¦ ÀÌ¿ëÇÑ Áõ¸íÀº Example 4.3.1¿¡ ÀÖÀ½)
4.3 Bivariate Transformations
(X,Y)ÀÇ
ºÐÆ÷·ÎºÎÅÍ (U,V)=(g1(X,Y),g2(X,Y))ÀÇ ºÐÆ÷¸¦ ±¸ÇÏ´Â ¹æ¹ý¿¡
´ëÇØ ¾Ë¾Æº»´Ù.
- discrete
case: (4.3.1)
- continuous and one-to-one transformation case: (4.3.2)
- continuous and many-to-one transformation case: (4.3.6)
- [Q] What if we have only one function, say U=g(X,Y),
of interest?
- [Q] ÀÌ»êÇü È®·üº¯¼öÀÏ ¶§ (4.3.1)½ÄÀ» ¾²±â À§ÇØ 'ÀÏ´ëÀÏ
´ëÀÀ º¯È¯'À̶ó´Â °¡Á¤ÀÌ ÇÊ¿äÇÑ°¡?
- [Q] Example 4.3.6¿¡¼ V=|Y| ´ë½Å V=Y ·Î µÎ°í U=X/YÀÇ
marginal distributionÀ» ±¸Çغ¸ÀÚ.
4.4 Hierarchical Models and Mixture Distributions
- [Q]
Hierarchical modelÀ̶õ ¹«¾ùÀÌ¸ç ¿Ö ÇÊ¿äÇÑ°¡?
- Hierarchical
modelÀÌ ÀÚ¿¬Çö»óÀ» ¼³¸íÇÏ´Â µ¥¿¡ ´õ ÀûÇÕÇÒ ¶§°¡ ÀÖ´Ù (Bayesian view):
Example 4.4.1, Example 4.4.6
- Hierarchical
model·Î »ý°¢ÇÔÀ¸·Î½á ±â´ë°ª µîÀÇ °è»êÀ» ´õ ½±°Ô ÇÒ ¼ö ÀÖ´Ù: noncentral
chi-squared distribution in p.166
- Theorem
4.4.3: EX = E(E(X|Y))
(ÁÖÀÇ: EÀÇ Àǹ̰¡ °¢°¢ ´Ù¸§¿¡ ÁÖÀÇÇÏÀÚ)
- mixture
distribution: Definition 4.4.4
Note: Let
f(y)=0.5f1(y) + 0.5f2(y), where f1(y)
is pdf of N(¥ì1,¥ò12), f2(y)
is pdf of N(¥ì2,¥ò22).
Then
f(y) is also a mixture distribution. (Why? [A] Y|X=i ¡ fi(y), P(X=1)=P(X=2)=1/2)
- Theorem 4.4.7: VarX = E(Var(X|Y)) + Var(E(X|Y)) (ÁÖÀÇ:
Á¤¸®ÀÇ Áõ¸íÀ» ÅëÇØ E(Var(X|Y))¿Í Var(E(X|Y))°¡ °¢°¢ ¹«¾ùÀ» ÀǹÌÇÏ´ÂÁö¸¦
Á¤È®È÷ ÀÌÇØÇÏ°í ÀÌ Á¤¸®¸¦ ¾Ï±âÇصÎÀÚ.)
- Poisson-gamma
mixture¿Í over-dispersion problem: [Q] Y~Poisson(¥ë) (´Ü,¥ë=E(¥Ë)=¥á¥â)ÀÏ ¶§¿Í Y|¥Ë~Poisson(¥Ë),
¥Ë~gamma(¥á,¥â) ÀÏ ¶§, Var(Y)¸¦ °¢°¢ ±¸ÇÏ¿© ºñ±³Çغ¸ÀÚ. (Âü°í: Example 4.4.5,
Exercise 4.32a. Poisson-gamma mixture¿¡¼¿Í °°ÀÌ Æ÷¾Æ¼Û ºÐÆ÷¿¡¼ Var(Y)
> EYÀÎ °æ¿ì '°ú»êÆ÷(over-dispersion) Çö»óÀÌ Á¸Àç'ÇÑ´Ù°í ÇÑ´Ù.
4.5 Covariance and Correlation
- Theorem 4.5.5: independence implies zero covariance.
But the converse is not true.
- Theorem 4.5.6: [Q] Var(¥ÒaiXi)=? or,
more generally, Cov(¥ÒaiXi, ¥ÒbjYj)=? (Âü°í:
Var(¥ÒaiXi)=Cov(¥ÒaiXi,¥ÒaiXi) )
- Theorem 4.5.7: The proof using Cauchy-Schwarz inequality
is recommended rather than the proof in p.172-3.
- The correlation coefficient indicates a linear relationship between X and Y.: Example 4.5.9
- Bivariate normal:
Definition 4.5.10¿¡¼´Â pdf·Î½á À̺¯·® Á¤±ÔºÐÆ÷¸¦ Á¤ÀÇÇÏ¿´´Ù. Exercise 4.46¿¡¼´Â
À̺¯·® Ç¥ÁØÁ¤±ÔºÐÆ÷ È®·üº¤Å͸¦ ÀÌ¿ëÇÏ¿© À̺¯·® Á¤±ÔºÐÆ÷¸¦ Á¤ÀÇÇÏ¿´´Ù.(representational
definition of the bivariate normal distribution). º¸´Ù ÀϹÝÀûÀ¸·Î ´Ùº¯·®
Á¤±ÔºÐÆ÷¸¦ ´ÙÀ½°ú °°ÀÌ Á¤ÀÇÇÒ ¼ö ÀÖ´Ù;
X ~ Nk(¥ì,¥Ò) ¢¢ X = AZ +
¥ì , where Z ~ Nk(0,I),
and A is nonsingular with AA'=¥Ò. (A can be represented as ¥Ò1/2 (why?))
- Properties of the bivariate normal distribution: a-d
in p.175, and the next one for the conditional distribution.
- (X,Y)'~N2((¥ìX ¥ìY)', ¥Ò), ¥Ò=
ÀÏ ¶§, Y|x ~N(¥ìY +¥ñ(¥òY /¥òX)(x-¥ìX),¥ò2Y
(1-¥ñ2)).
- Marginal normality does not imply joint normality. (cf.
Exercise 4.47)
4.6 Multivariate Distributions
- µÎ È®·üº¯¼ö¿¡ °üÇÑ ¼ºÁúÀ» n°³ÀÇ È®·üº¯¼ö¿¡ °üÇÑ ¼ºÁú·Î
È®Àå (Á¤¸® 4.6.6, 4.6.7, 4.6.11, 4.6.12)
- Multinomial distribution:
- If (X1, X2,
... ,Xm)~Multinomial(n,p=(p1,p2,....,pm)), where X1+X2+
... +Xm=n, p1+p2+ ...+pm=1,
then Xi ~ b(n,pi)
- If (X1, X2,
... ,Xm)~Multinomial(n,p), then (X2, ... ,Xm)|X1~Multinomial(n-X1, p=(p2/(1-p1),p3/(1-p1),....,pm/(1-p1)))
- [Q] What is the conditional distribution of (X3, ... ,Xm) given (X1,X2)?
- mutual independence¿Í pairwise independenceÀÇ °ü°è (Âü°í:[Q]
mutually independent sets¿Í mutually exclusive(or disjoint) setsÀÇ ±¸º°;
Definition 1.1.5, Exercise 1.39 ÂüÁ¶)
- Example 4.6.13: distributions of order statistics of
exponential distribution and the spacings;
- Let X1, X2,
... ,Xn be a
random sample from Exponential(¥ë), and Y1, Y2,
... ,Yn be the
order statistics. Then the normalized spacings nY1, (n-1)(Y2-Y1), ..., 2(Yn-1-Yn-2),
Yn-Yn-1 are iid Exponential(¥ë).
[Q] give a proof of this.
4.7 & 3.6 Inequalities and Identities
- Hölder's inequality: Cauchy-Schwarz inequality and
Liapounov's inequality are special cases. Also see (4.7.9).
- Minkowski's inequality
- Jensen's inequality
- covariance inequality (Theorem 4.7.9)
- Chebyshev's inequality p.122 and Markov's inequality
in p.136 ([Q] What's the difference? [Q] give a proof of Markov's inequality.)
- Example 3.6.2: P(|X-¥ì|¡Ãt¥ò) ¡Â 1/t2. ÀÌ ºÎµî½ÄÀº ±â´ë°ª°ú ºÐ»êÀÌ Á¸ÀçÇÏ´Â ¸ðµç È®·üº¯¼ö¿¡ Àû¿ëÇÒ
¼ö ÀÖ´Ù´Â ÀåÁ¡ÀÌ ÀÖÀ¸³ª ³Ê¹« ¹«µò(»óÇÑ°ª 1/t2ÀÌ ³Ê¹« Å«) ´ÜÁ¡µµ ÀÖ´Ù. Example 3.6.3°ú 136ÂÊ Theorem 3.8.2 ÂüÁ¶.
- 3.6Àý: recursion relations
- Stein's Lemma (Lemma 3.6.5): [Q] What's the use of this
lemma?
Á¦4Àå (3.6Àý) ¿¬½À¹®Á¦(¼÷Á¦)
5 6 7 11 12 13 14
15 17 19 21 24 26 27 28
31 32 36 39 41 45 47
52 53 54 55 56 58 64 3.44 3.46 3.49(b)