{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture9 - Stochastic Calculus The Normal Distribution...

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
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
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 4
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
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern