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lecture9

# lecture9 - Stochastic Calculus The Normal Distribution...

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Stochastic Calculus The Normal Distribution Preliminaries: Normal Random Variables Definition: A random variable Z with values in R is said to be normally distributed with mean μ and variance σ 2 > 0 if Z has density p ( z ) = 1 σ 2 π e ( z μ ) 2 / 2 σ 2 . In this case, we write Z N ( μ, σ 2 ). Z is said to be a standard normal RV if Z N (0 , 1). Remark: As σ 2 0, the distribution of Z concentrates on the point μ . Thus we say that Z is a degenerate normal RV with mean μ and variance 0 and write Z N ( μ, 0) if P ( Z = μ ) = 1 . Jay Taylor (ASU) APM 530 - Lecture 9 Fall 2010 1 / 42

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Stochastic Calculus The Normal Distribution Properties of Normal Random Variables A ne transformations of normal RVs are normal: if Z N ( μ, σ 2 ) and W = aZ + b , then W N ( a μ + b , a 2 σ 2 ) Sums of independent normal RVs are normally distributed: if Z 1 and Z 2 are independent and Z i N ( μ i , σ 2 i ), then Z 1 + Z 2 N ( μ 1 + μ 2 , σ 2 1 + σ 2 2 ) . Jay Taylor (ASU) APM 530 - Lecture 9 Fall 2010 2 / 42
Stochastic Calculus The Normal Distribution Multivariate Normal Random Variables Definition: A random variable Z = ( Z 1 , · · · , Z d ) with values in R d is said to be normally distributed if for every vector b R d , the real-valued random variable W = b · Z = b 1 Z 1 + · · · + b d Z d is normally distributed. The distribution of Z is completely determined by its mean μ = ( μ 1 , · · · , μ d ) and covariance matrix Σ = ( Σ ij ), where μ i = E Z i and Σ ij = Cov ( Z i , Z j ) , and we write Z N ( μ, Σ ). Jay Taylor (ASU) APM 530 - Lecture 9 Fall 2010 3 / 42

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Stochastic Calculus The Normal Distribution Properties of Multivariate Normal Random Variables If Σ is non-singular, then Z has a density p ( z ) = 1 (2 π ) d det( Σ ) exp 1 2 ( z μ ) T Σ 1 ( z μ ) . If Z N ( μ, Σ ) and W = AZ + B , where A R n × d and B R n , then W N ( A μ + B , A Σ A T ) If Z 1 and Z 2 are independent and Z i N ( μ i , Σ i ), then Z 1 + Z 2 N ( μ 1 + μ 2 , Σ 1 + Σ 2 ) . Jay Taylor (ASU) APM 530 - Lecture 9 Fall 2010 4 / 42
Stochastic Calculus The Normal Distribution The Central Limit Theorem The following theorem explains why the normal distribution is of special interest. Theorem: Suppose that X 1 , X 2 , · · · are independent real-valued RVs with E [ X i ] = μ and Var( X i ) = σ 2 , and let S n = X 1 + · · · + X n . Then, for every z R , lim n →∞ P S n n μ σ n z = P ( Z z ) , where Z N (0 , 1) is a standard normal random variable. Jay Taylor (ASU) APM 530 - Lecture 9 Fall 2010 5 / 42

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Stochastic Calculus The Normal Distribution Convergence in Distribution Definition: Suppose that Z and Z 1 , Z 2 , · · · are real-valued RVs with cumulative distribution functions (cdf’s) F n ( x ) = P ( Z n x ) and F ( x ) = P ( Z x ) . We say that the sequence ( Z n ) 1 converges in distribution to Z and write Z n Z if F n ( x ) F ( x ) as n → ∞ at every point x at which F is continuous.
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