c, c ∈ R: ˚Ñb(constant) ƒb, w2 c ªJÑL<õb, ƒb f (x) .}Äщb x 5Z‰7

É  !€bܵ3

I. àƒbÜ

ƒb(function) àVH[ç/_‰b (value) 5MZ‰v, ª?ßÞú“/ᔥ@5Móú@5Zb, à f (x)g(x)¯U VH[ç‰b x 5ƒb
Ê$l2àíƒbÜà-:

(1) f (x) = c, c ∈ R: ˚Ñb(constant) ƒb, w2 c ªJÑL<õb, ƒb f (x) .}Äщb x 5Z‰7‰

(2) f (x) = anxⁿ+ an−1xⁿ⁻¹+ · · · + a1x+ a0, w2 ai∈ R, i = 1, 2, . . . , n,n ∈ N :f (x) ˚ÑÖá (polynomial)

Ñ|c5ƒb

(3) f (x) = e^g(x): Nbƒb
/ e⁰= 1,e^−∞= _e^1∞ = 0

(4) f (x) = logag(x): úbƒb (log function), ÑJ a Ñ5úbƒb, w2 a Ñøb (constant)
|c5úbƒ bÑJAÍb e Ñ5ƒb, J f (x) = ln g(x) [ý
I mn Ñs_b, úbƒbxJ-Ô4:

(i) log_amn= log_am+ log_an (ii) log_a^m_n = log_am− log_an (iii) log_amⁿ= n log_am (iv) log_a1 = 0

(v) log_aa= 1 (vi) x = e^{ln x}

II.  }

 }Ñ|à5bçj¶, Ê$l2:

} àk°ƒb5”M, Ê$l2àV°j©/Óœ‰b5VbêÞ5FÊ; £‚à|×–N¶

(maximum likelihood) V°j¡b|ª?êÞ5M

} àk°jƒbF¨Ö5Þ £ñ 5M, Ê$l2àV°j©/Óœ‰bÊ/¨–ÈêÞ5œ0(probability)

Ìb (mean) C‚M (expected value) £‰æb (variance)  Ê$l2àí }tà-:

f(x) = c, c ∈ R f^′(x) = 0 R

f(x)dx = cx + d

f(x) = cx, c ∈ R f^′(x) = c R

f(x)dx = ¹₂cx²+ d f(x) = cxⁿ, c ∈ R, n ∈ N f^′(x) = cnxⁿ⁻¹ R

f(x) = _n+1¹ cxⁿ⁺¹+ d

f(x) = e^x f^′(x) = e^x R

f(x)dx = e^x+ d

f(x) = e^cx f(x) = ce^cx R

f(x)dx = ¹_ce^cx+ d f(x) = ln x, x > 0, f^′(x) = ¹_x

f(x) = ¹_x R

f(x)dx = ln |x| + d f(x) = ln g(x), g(x) > 0, f^′(x) = ^g

′(x) g(x)

f(x) = cg(x) f^′(x) = cg^′(x) R

f(x)dx = cR g(x)dx f(x) = g(x) ± h(x) f^′(x) = g^′(x) ± h^′(x) R

f(x)dx =R

g(x)dx ±R h(x)dx f(x) = g(x) · h(x) f^′(x) = g^′(x)h(x) + g(x)h^′(x)

wFàí }t:

(1) ©ā¶† (chain rule): ^dy_dx = ^dy_du^du_dx (2) ¶} }¶ (integration by parts): R

udv= uv −R vdu

Ex 1. I f (x) = _b−a¹ , w2 ab Ñs_b (constant), / b > a, ° (a) Rt

af(x)dx; (b) Rb

a xf(x) dx; (c) Rb

ax²f(x)dx (Sol.)

Note: Êœ02 f (x) = _b−a¹ Ñk a D b 5È5ÌG‰b (uniform random variable) 5œ0}ºƒb (probability distribution function),

• â (a) üæªø, J t ∈ [a, b], †Ú œ0ƒbÑP(X ≤ t) = ^t−a_b−a

• â (b) üæªøRb

a xf(x)dx = E(X), FJÌG‰b5‚MÑ E(X) = ^a+b₂

• â (c) üæªø, ÌGÓœ‰b5‰æb V (X) = E(X²) − [E(X)]²=¹₃(b²+ ab + a²) − ^a+b₂
2

=^(a−b)₁₂ ²

Ex 2. I f (x) = λe^−λxdx, w2 λ > 0 Ñøb, t° (a)Rt

0f(x)dx; (b)R^∞

0 xf(x)dx; (c)R^∞

0 x²f(x)dx (Sol.)

Note: Êœ02 f (x) = λe^−λX,λ > 0 / x > 0, Ñ¡bÑ λ 5NbÓœ‰b (exponential random variable) 5œ0 }ºƒb (probability distribution function),

• â (a) üæªø, J t ∈ [a, b], †Ú œ0ƒbÑP(X ≤ t) = 1 − e^−λt

• â (b) üæªøRb

a xf(x)dx = E(X), FJÌG‰b5‚MÑ E(X) = ¹_λ

• â (c) üæªø, ÌGÓœ‰b5‰æb V (X) = E(X²) − [E(X)]²=_λ²2 − _λ¹
2

=_λ¹2

III. ,¸¯U

I a1, a2, . . . , an Ñøb, Êbç2‚à,¸¯U (P) VH[©/ n á5,¸, c5,¸tà-:

(1) I ai= c, w2 c Ñøb, † Pn i=1

ai= c + c + · · · + c

| {z }

n _ c ó‹

= n · c

(2) I ai= i, † Pn i=1

= 1 + 2 + · · · + n = ⁿ⁽ⁿ⁺¹⁾₂

(3) I ai= i², † Pn i=1

= 1²+ 2²+ · · · + n²=n(n+1)(2n+1) 6

(4) I ai= a1rⁱ⁻¹, w2 a1∈ R, † a1, a2,· · · , an Ñø̪b, †¤Ìªb5¸

Sn= Pn i=1

= a1+ a1r+ a1r²+ · · · + a1rⁿ⁻¹= a1(1 + r + · · · rⁿ⁻¹) = ^a1(1−r

n) 1−r

(5) I ai= a1rⁱ⁻¹, w2 a1∈ R / r < 1 † a1, a2, a3,· · · , ÑøÌ̪b, †¤Ì̪b5¸

S= P∞ i=1

= a1+ a1r+ a1r²+ · · · = a1(1 + r + r²+ · · · ) = _1−r^a1

IV. lbj¶

Theorem ( ¶¶†)

JøK T.Û%âs_¥ VêA, ø_¥ ªŽâ m .°íj¶VêA, ù_¥  n .°íj¶VêA, †u ªJ m × n .°íj¶VêAv T

Theorem (‹¶¶†)

JøK TªJ‚às.°íj¶VêA, à‹ø_j¶ m j¶VêA, ù_j¶ n .°íj¶VêA, †u ªJ m + n .°íj¶VêA¤ T

Theorem

øÓñßå4í§¶˚ѧ (permutation)

(1) n _.°Óñ§íj¶ n! , w2 n! = n(n − 1)(n − 2) · · · 1

(2) * n _.°íÓñ2¦ r _ÓñV§, u P_rⁿ C_nPr, w2 Prⁿ =_(n−r)!^n!
(3) * n _.°íÓñ2¦ r _ÓñV§, JcqªJ½º²¦, †u n^r j¶

(4) n _Óñ2 k _Óñuó°í, †§j¶ ^n!_k! 

(5) n _Óñ, w2 k1_Óñuó°í,k2_Óñuó°í,. . ., ks_Óñuó°í, †ø n _ÓñV§u_k ^n!

1!k₂!···ks!
Theorem

øÓñd³ßå4í§¶˚Ñ ¯ (combination)

(1) * n _.°íÓñ2¦ r _Óñ$Aø ¯íj¶ C_rⁿ C_nCr , w2 Crⁿ= _r!(n−r)!^n!

(2) * n _.°íÓñ2¦ n − r _Óñ$Aø ¯íj¶ Cn−rⁿ C _n−rCr, w2 Cn−rⁿ = _r!(n−r)!^n! = Crⁿ

Ex 3. ¬Ç, â A ƒ B •¡ (Ér →, ↑), t°

(1) â A ƒ B r¶í•¶?

(2) â A ƒ B .%¬ C 핶?

(3) B* A •ƒ B, ~½B}%¬ C õíœ0Öý?

(Sol.)

Ex 4. An “all possible regressions” search of a data set containing 7 independent variables will produce . (93 «× F)

(A) 13 regressions (B) 48 regressions (C) 64 regressions (D) 127 regressions (E) none of the above

(Sol.)

Ex 5. øû.°ízJpú_.°íz*2, ©øz*.âz/*25z.ª¹¯, †uJ¶?

(Sol.)

Ex 6. t½å‚ PEPPER Öý.°í§?

(Sol.)

Ex 7. * 20 A2²¦ 3 A$Aøãº}, t½ A.°ãº}íj¶Öý?

(Sol.)

Ex 8. Of 10 people on a student newspaper staff, 6 are familiar with a particular word processing package. If three people are randomly selected as assigned to work as a team, how many ways could a team be selected in which one was familiar with the word processing package and the other two were not? (92 «× F)

(A) 24 (B) 120 (C) 20 (D) 36

(Sol.)

Theorem (Binomial Theorem)

(x + y)ⁿ= Xn i=1

C_iⁿxⁱyⁿ⁻ⁱ

(ax + by)ⁿ= Xn i=1

C_iⁿ(ax)ⁱ(by)ⁿ⁻ⁱ

Ex 9. ~½-Ç2,x³y⁴ í[bÑS?

(1) (x + y)⁷ (2) (3x + 4y)⁷

(Sol.)