同济大学：《概率论与数理统计》课程教学资源（讲稿）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(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

8.2 CHEBYSHEV'S INEQUALITY AND THE WEAK LAW 8.3 The Central Limit Theorem Theorem 2.1 The weak law of large numbers Let X1,...,Xn be a sequence of independent and identically distributed random variables.Each having finite mean E(Xi)u,D(Xi)-2.Then,for any c>0, P+,+X-川≥)→0 asn→∞ 4口5,42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 Theorem 2.1 The weak law of large numbers Let X1, · · · , Xn be a sequence of independent and identically distributed random variables. Each having finite mean E(Xi) ˆ=µßD(Xi) ˆ=σ 2 . Then, for any  > 0, P(| X1 + · · · + Xn n − µ| ≥ ) → 0 as n → ∞ Xiaohan Yang Chapter 8 Limit Theorems

8.2 CHEBYSHEV'S INEQUALITY AND THE WEAK LAW 8.3 The Central Limit Theorem Theorem3.1 The Central Limit Theorem Let X1,...,X be a sequence of independent and identically distributed random variables.Each having finite mean E(Xi)=u and variance D(Xi)=o2.Then the distribution of X+…+Xn-n4 aVn tends to the standard normal as noo.That is,for -00<a<十0， imP2X-业≤X刘=() n-0o Vna 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 Theorem3.1 The Central Limit Theorem Let X1, · · · , Xn be a sequence of independent and identically distributed random variables. Each having finite mean E(Xi) = µ and variance D(Xi) = σ 2 . Then the distribution of X1 + · · · + Xn − nµ σ √ n tends to the standard normal as n → ∞. That is, for −∞ < a < +∞ß lim n→∞ P( Pn i=1 Xi − nµ √ nσ ≤ x) = Φ(x) Xiaohan Yang Chapter 8 Limit Theorems

8.2 CHEBYSHEV'S INEQUALITY AND THE WEAK LAW 8.3 The Central Limit Theorem EXAMPLE 3C If 10 fair dice are rolled,find the approximate probability that the sum obtained is between 30 and 40,inclusive. 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 EXAMPLE 3C If 10 fair dice are rolled, find the approximate probability that the sum obtained is between 30 and 40, inclusive. Xiaohan Yang Chapter 8 Limit Theorems

8.2 CHEBYSHEV'S INEQUALITY AND THE WEAK LAW 8.3 The Central Limit Theorem Theorem3.3 De Moivre-Laplace The Central Limit Theorem Let X1,...,Xn be a sequence of independent and identically distributed random variables.Each Xi B(1,p).Then H-00<X<+00, mPX-吧≤x)=(x) Vnp(1-p) 1Xi~B(n,p),.when Y~B(n,p),n is large, P(a<X≤b)= ∑cpP1-p)- a<k≤b 中( b-np a-np √np(1-p) Vnp(1-p) 0a=0时，认为x=一0： b=n时，认地.9 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 Theorem3.3 De Moivre-Laplace The Central Limit Theorem Let X1, · · · , Xn be a sequence of independent and identically distributed random variables. Each Xi ∼ B(1, p). Then ∀ − ∞ < x < +∞ß lim n→∞ P( Pn i=1 Xi − np p np(1 − p) ≤ x) = Φ(x) 1 ∵ Pn i=1 Xi ∼ B(n, p)ß∴ when Y ∼ B(n, p)ßn is largeß P(a < X ≤ b) = X a<k≤b C k n p p (1 − p) n−k ≈ Φ( b − np p np(1 − p) ) − Φ( a − np p np(1 − p) ) 2 a = 0ûß@èx = −∞¶b = nûß@èb = +∞ Xiaohan Yang Chapter 8 Limit Theorems

8.2 CHEBYSHEVS INEQUALITY AND THE WEAK LAW 8.3 The Central Limit Theorem Theorem3.3 De Moivre-Laplace The Central Limit Theorem Let X1,...,Xn be a sequence of independent and identically distributed random variables.Each Xi~B(1,p).Then V-00<X<十00， lim P(ZE1X-mp n→ Vnp(1-p) ≤x)=(x) .i B(n,p),.when Y~B(n,p),n is large, P(a<X≤b)=∑ChpP(1-p)-k a<k<b b-np -lm-可 a-np mp(1-p) 。a=0时，认为x=-∞；b=n时，认为地=才0：，，至a。 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 Theorem3.3 De Moivre-Laplace The Central Limit Theorem Let X1, · · · , Xn be a sequence of independent and identically distributed random variables. Each Xi ∼ B(1, p). Then ∀ − ∞ < x < +∞ß lim n→∞ P( Pn i=1 Xi − np p np(1 − p) ≤ x) = Φ(x) 1 ∵ Pn i=1 Xi ∼ B(n, p)ß∴ when Y ∼ B(n, p)ßn is largeß P(a < X ≤ b) = X a<k≤b C k n p p (1 − p) n−k ≈ Φ( b − np p np(1 − p) ) − Φ( a − np p np(1 − p) ) 2 a = 0ûß@èx = −∞¶b = nûß@èb = +∞ Xiaohan Yang Chapter 8 Limit Theorems