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λ€λ³μ λΆν¬μ λν΄ μμλ³΄κΈ° μ μ, λͺκ°μ§ μ€μν λΆλ±μμ λ³΄μλλ€. μ λ¦¬ : μ  μΌ λΆλ±μ (Jensen Inequality)
νλ₯ λ³μ $X$μ λ³Όλ‘ν¨μ $\phi$μ λνμ¬, λ€μμ΄ μ±λ¦½νλ€. $$\phi(\E(X)) \leq \E(\phi(X))$$
$\E$λ₯Ό (νλ₯ μ λ°μν) κ°μ€ νκ· μΌλ‘ μκ°νλ©΄, λ³Όλ‘ν¨μμ λν΄ (κ°μ€ νκ· μ ν©μκ°) μ (ν¨μκ°μ κ°μ€ νκ· ) μ΄νλΌλ μ λ¦¬μλλ€.

μ΄κ³ λ―ΈλΆμ μ΄μ©νλ©΄ μ΄ λΆλ±μμ μ½κ² μ¦λͺν  μ μμ§λ§, $\phi$κ° λλ² λ―ΈλΆκ°λ₯ν ν¨μμΌ λλ§ κ°λ₯ν μ¦λͺμ΄λΌλ νκ³κ° μμ΅λλ€. μ¬κΈ°μλ μ‘°κΈ λ€λ₯Έ μ¦λͺμ μ΄μ©νκ² μ΅λλ€. μμ λ€μ μ°λ©΄, λ³Όλ‘ν¨μ $\phi$μ λν΄, λ€μμ λ³΄μ΄λ©΄ μΆ©λΆν©λλ€. $$\phi\left(\int_{A} g(x) \dd{x}\right) \leq \int_{A} \phi(g(x)) \dd{x}$$ λ³Όλ‘ν¨μμ μ μμ λ°λΌ, μμμ $x_0$ μ λν΄, $ax + b \leq \phi(x)$, $ax_0 + b = \phi(x_0)$ λ₯Ό λ§μ‘±νλ μ΄λ€ μ§μ  $y = ax + b$κ° μ‘΄μ¬ν©λλ€. (μ΄λ₯Ό $\phi$μ sub-derivativeλΌ νλλ°, $\phi$κ° μ°μμΈ λ³Όλ‘ν¨μκΈ°λ§ νλ©΄ λ―ΈλΆκ°λ₯νμ§ μμλ μ‘΄μ¬ν©λλ€. μ§κ΄μ μΌλ‘ λ³΄μ΄κΈ°λ λ³λ‘ μ΄λ ΅μ§ μμΌλ, (μ μ΄λ μ κ° μλ μ¦λͺμ) supporting hyperplane theoremμ΄λΌλ μλΉν κ°ν ν΄μ μκ΅¬ν©λλ€. μ¬κΈ°μλ μΌλ³μλ§ λ³Ό κ²μ΄λ―λ‘ μ¦λͺ μλ΅.)

$x_0 = \displaystyle \int_{A} g(x) \dd{x}$λΌ νκ³  μ΄λ₯Ό κ·Έλλ‘ νμ©νλ©΄, λ€μμ΄ μ±λ¦½ν©λλ€. $$\int_{A} \phi(g(x)) \dd{x} \geq \int_{A} ag(x) + b \dd{x} \geq a \int_{A} g(x) \dd{x} + b = ax_0 + b = \phi\left(\int_{A} g(x) \dd{x}\right)$$

μ λ¦¬ : λ¦¬μΌνΈλΈν λΆλ±μ (Liapounov Inequality)
νλ₯ λ³μ $X$μ λν΄ $\E(\abs{X}^s) < \infty$ μ΄λ©΄, $0 < r < s$μΈ $r$μ λν΄, λ€μμ΄ μ±λ¦½νλ€. $$\E(\abs{X}^r)^{1/r} \leq \E(\abs{X}^s)^{1/s}$$

μ¦λͺ : $p > 1$μ λν΄ $\phi(x) = \abs{x}^p$ λ‘ μ  μΌ λΆλ±μμ μ°λ©΄ $\E(\abs{X}^p) \leq \E(\abs{X})^p$ μλλ€.
$p = s / r > 1$ κ³Ό νλ₯ λ³μ $\abs{X}^r$ λ₯Ό μ΄ μ  μΌ λΆλ±μμ λμνλ©΄ μ¦λͺ λ.

μ λ¦¬ : λ§λ₯΄μ½ν λΆλ±μ (Markov Inequality)
νλ₯ λ³μ $X$μ λν΄ $\E(\abs{X}^r) < \infty$ μ΄λ©΄, μμμ $k$μ λν΄ λ€μμ΄ μ±λ¦½νλ€. $$\P(\abs{X} \geq k) \leq \E(\abs{X}^r) / k^r$$

μ¦λͺ : $\P(\abs{X} \geq k) = \E(I_{(\abs{X} \geq k)})$ λ‘ μλλ€. (indicator function) μ΄λ, $I_{(\abs{X} \geq k)}$μ λν΄ μκ°ν΄ λ³΄λ©΄ μ΄λ λ€μ $I_{(\abs{X} / k \geq 1)}$ κ³Ό κ°κ³ , μ΄ ν¨μλ $\abs{X} / k$ κ° 1λ³΄λ€ ν° λΆλΆμμλ§ 1μ΄λ―λ‘ λ€μμ΄ μ±λ¦½ν©λλ€. $$I_{(\abs{X} / k \geq 1)} \leq (\abs{X} / k)^r I_{(\abs{X} / k \geq 1)}$$ μλ³μ κΈ°λκ°μ μμ°λ©΄ μ£Όμ΄μ§ μμ΄ λ©λλ€.

μ λ¦¬ : μ²΄λΉμ°ν λΆλ±μ (Chebyshev Inequality)
νλ₯ λ³μ $X$μ λν΄ $\V(X) < \infty$ μ΄λ©΄, μμμ $k$μ λν΄ λ€μμ΄ μ±λ¦½νλ€. $$\P(\abs{X - E(X)} \geq k) \leq \V(X) / k^2$$

μ¦λͺ : μμ  Markov λΆλ±μμμ, $Z = X - E(X)$, $r = 2$λ₯Ό λμν©λλ€.