ENGG2470A.Sp.19.lec.07.post-class.pdf - ENGG2470A Week 8 Example Discrete RV Application Example and Continuous RV Minghua Chen([email protected]

# ENGG2470A.Sp.19.lec.07.post-class.pdf - ENGG2470A Week 8...

• 20

This preview shows page 1 - 7 out of 20 pages.

ENGG2470A Week 8: Example Discrete RV, Application Example, and Continuous RV Minghua Chen ([email protected]) Information Engineering The Chinese University of Hong Kong Reading: Ch. 2.5-2.8 and 3.1-3.2 of the textbook M. Chen ENGG2470A Week 8 1 / 20
Expectation of the Sum Let X 1 ,... X n be n random variables with finite E [ X i ] (1 i n ) (not necessary independent), then E " n i = 1 X i # = n i = 1 E [ X i ] Example : X 1 and X 2 are two Bernoulli random variables with parameter 1 / 2, and X 1 + X 2 = 1. Then E [ X 1 + X 2 ] = ( 1 + 0 ) · 0 . 5 +( 0 + 1 ) · 0 . 5 = 1 = E [ X 1 ]+ E [ X 2 ] I But Var ( X 1 + X 2 ) = 0 6 = Var ( X 1 )+ Var ( X 2 ) = 1 4 + 1 4 = 1 2 M. Chen ENGG2470A Week 8 2 / 20
Example: The Umbrella Problem n students go to a dinning hall for lunch. Each throws an umbrella into a box. When leaving, each of them randomly takes one umbrella. I Define a r.v. Y as number of students who get their own umbrellas. I E [ Y ] =? Define Bernoulli (indicator) r.v.s X i = ( 1 , if the ith student picks own umbrella; 0 , otherwise. I p X i ( 1 ) =? 1 n . (By counting: ( n - 1 )! / n ! ) I E [ X i ] = 1 n . I Thus we have Y = n i = 1 X i , and E [ Y ] = n i = 1 E [ X i ] = 1 . M. Chen ENGG2470A Week 8 3 / 20
The Umbrella Problem The variance of Y can be computed by Var ( Y ) = E Y 2 - E 2 [ Y ] = E Y 2 - 1 . Y 2 = n i = 1 X 2 i + i , j : i 6 = j X i X j E X 2 i = 1 n P ( X i X j = 1 ) = 1 n 1 n - 1 Var ( Y ) = E Y 2 - 1 = n i = 1 E X 2 i + i , j : i 6 = j E [ X i X j ] - 1 = 1 + 1 - 1 = 1 . M. Chen ENGG2470A Week 8 4 / 20
Review: Binomial RV Experiment : flip a coin independently for n times. Define a Binomial r.v. Y be the total number of HEADs we obtained. Y = n i = 1 X i I X i , ( i = 1 , 2 ,..., n ) : Bernoulli r.v.s modeling the outcome of the i -th flip I The PMF of Binomial r.v. Y is given by p Y ( k ) = P ( Y = k ) = P n i = 1 X i = k ! = ( n k ) p k ( 1 - p ) n - k . I The expectation and variance of Y E [ Y ] = n i = 1 E [ X i ] = np , Var ( Y ) = n i = 1 Var ( X i ) = np ( 1 - p ) M. Chen ENGG2470A Week 8 5 / 20
Poisson Random Variable X is a Poisson random variable if its PMF has the form of P X ( k ) = e - λ λ k k !

#### You've reached the end of your free preview.

Want to read all 20 pages?

• Winter '20

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern