8.2 CHEBYSHEVS INEQUALITY AND THE WEAK LAW( 8.3 The Central Limit Theorem 3o rule of normal distribution Suppose X N(u,o2) P(X-4<ka)=P(<k)=2Φ(k)-1 0P(0X-4<o)=2Φ(1)-1=0.683 0P(X-4<2a)=2Φ(2)-1=0.955 0P(X-4<3G)=2Φ(3)-1=0.997 4口t5,48)元2月00 Xiaohan Yang Chapter 8 Limit Theorems
logo 8.2 CHEBYSHEV’S INEQUALITY AND THE WEAK LAW OF LARGE NUMBERS 8.3 The Central Limit Theorem 3σ rule of normal distribution Suppose X N(µ, σ2 ) P(|X − µ| < kσ) = P(| x − µ σ | < k) = 2Φ(k) − 1 1 P(|X − µ| < σ) = 2Φ(1) − 1 = 0.683 2 P(|X − µ| < 2σ) = 2Φ(2) − 1 = 0.955 3 P(|X − µ| < 3σ) = 2Φ(3) − 1 = 0.997 Xiaohan Yang Chapter 8 Limit Theorems
8.2 CHEBYSHEV'S INEQUALITY AND THE WEAK LAW 8.3 The Central Limit Theorem 30 rule of normal distribution Suppose X N(u,o2) PIX-4<ko)=PI2,1<k)=2(k)-1 0P(X-川<o)=2Φ(1)-1=0.683 gP(X-4川<2a)=2φ(2)-1=0.955 0P(X-4<3a)=2Φ(3)-1=0.997 4日5,43)手，3月00 Xiaohan Yang Chapter 8 Limit Theorems
logo 8.2 CHEBYSHEV’S INEQUALITY AND THE WEAK LAW OF LARGE NUMBERS 8.3 The Central Limit Theorem 3σ rule of normal distribution Suppose X N(µ, σ2 ) P(|X − µ| < kσ) = P(| x − µ σ | < k) = 2Φ(k) − 1 1 P(|X − µ| < σ) = 2Φ(1) − 1 = 0.683 2 P(|X − µ| < 2σ) = 2Φ(2) − 1 = 0.955 3 P(|X − µ| < 3σ) = 2Φ(3) − 1 = 0.997 Xiaohan Yang Chapter 8 Limit Theorems
8.2 CHEBYSHEVS INEQUALITY AND THE WEAK LAW( 8.3 The Central Limit Theorem 3o rule of normal distribution Suppose X N(u,o2) P(X-4<ka)=P(<k)=2Φ(k)-1 0P(IX-4<o)=2φ(1)-1=0.683 0P(IX-4<2a)=2(2)-1=0.955 0P(IX-4<3o)=2φ(3)-1=0.997 4口913)元，王000 Xiaohan Yang Chapter 8 Limit Theorems
logo 8.2 CHEBYSHEV’S INEQUALITY AND THE WEAK LAW OF LARGE NUMBERS 8.3 The Central Limit Theorem 3σ rule of normal distribution Suppose X N(µ, σ2 ) P(|X − µ| < kσ) = P(| x − µ σ | < k) = 2Φ(k) − 1 1 P(|X − µ| < σ) = 2Φ(1) − 1 = 0.683 2 P(|X − µ| < 2σ) = 2Φ(2) − 1 = 0.955 3 P(|X − µ| < 3σ) = 2Φ(3) − 1 = 0.997 Xiaohan Yang Chapter 8 Limit Theorems
8.2 CHEBYSHEV'S INEQUALITY AND THE WEAK LAW 8.3 The Central Limit Theorem Markov's inequality If X is a random variable that takes only nonnegative values,then for any value a >0, P(X≥a)≤ E[X] 日5,421手，3000 Xiaohan Yang Chapter 8 Limit Theorems
logo 8.2 CHEBYSHEV’S INEQUALITY AND THE WEAK LAW OF LARGE NUMBERS 8.3 The Central Limit Theorem Markov’s inequality If X is a random variable that takes only nonnegative values, then for any value a > 0ß P(X ≥ a) ≤ E[X] a Xiaohan Yang Chapter 8 Limit Theorems
8.2 CHEBYSHEVS INEQUALITY AND THE WEAK LAW( 8.3 The Central Limit Theorem Chebyshev's's inequality If X is a random variable with finite mean E(X)=u and variance D(X)=a2,then for any valuee>0 P0X-川≥≤爱 Equivalently P(IX-E(X)1≥c)≤Dy 口15,18)4元，3000 Xiaohan Yang Chapter 8 Limit Theorems
logo 8.2 CHEBYSHEV’S INEQUALITY AND THE WEAK LAW OF LARGE NUMBERS 8.3 The Central Limit Theorem Chebyshev’s’s inequality If X is a random variable with finite mean E(X) = µ and variance D(X) = σ 2 , then for any value > 0 P(|X − µ| ≥ ) ≤ σ 2 2 Equivalently P(|X − E(X)| ≥ ) ≤ D(X) 2 Xiaohan Yang Chapter 8 Limit Theorems