stat1801_4

stat1801_4 - Uniform Distribution ( 29 <...

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Uniform Distribution ( 29 < <- = otherwise if 1 b x a a b x f f ( x ) a b ( 29 b a U X , ~ ( 29 -- < = b x b x a a b a x a x x F if 1 if if ( 29 2 a b X E + = ( 29 3 2 2 2 a ab b X E + + = ( 29 ( 29 12 2 a b X Var- = Uniform Distribution Example: Drop a needle on the floor Angle between needle tip and north direction ( 29 2 , ~ U ( 29 = + = 2 2 E ( 29 ( 29 3 12 2 2 2 =- = Var 8 1 4 2 2 1 4 2 2 4 = - = - = F F P Uniform Distribution Example: Buffons Needle Problem D D D L (< D ) ( 29 ? line a cross = P Drop Uniform Distribution Example: Buffons Needle Problem D ( 29 < = sin 2 line a cross L X P P X 2 , ~ D U X ( 29 , ~ U D L dxd D L 2 2 2 sin = = pD L 2 = Monte Carlo Uniform Distribution ( 29 1 , ~ U X 1 1 2- = X Y-1 1 X Y 2 = 2 ( 29 1 , 1 ~ 1 2-- = U X Y Uniform Distribution ( 29 6 , 2 ~- U X-2 6 X Y- = 6 8 8 6 X Y- = 1 ( 29 1 , ~ 8 6 U X Y- = X is said to have a binomial distribution with n trials and success probability p Binomial Distribution Bernoulli Experiment/Trial Possible outcome Probability Success p Fail 1 p H/T B/G A B C D E Correct / Wrong Sup / Opp Def / Non-def X = No. of successes in n independent Bernoulli trials ( 29 p n b X , ~ Binomial Distribution Value of X 1 2 3 4 Outcome FFFF SFFF FSFF FFSF FFFS SSFF SFSF SFFS FSSF FSFS FFSS SSSF SSFS SFSS FSSS SSSS Prob. (0.8) 4 (0.2)(0.8) 3 (0.2 )2 (0.8) 2 (0.2 )3 (0.8) (0.2) 4 No. of combinations 1 4 6 4 1 Example: n = 4, p = 0.2 4 C 4 C 1 4 C 2 4 C 3 4 C 4 ( 29 ( 29 ( 29 ( 29 4 , 3 , 2 , 1 , , 8 . 2 . 4 4 = = = =- x x x X P x p x x ( 29 4096 . = p ( 29 4096 . 1 = p ( 29 1536 . 2 = p ( 29 0256 . 3 = p ( 29 0016 . 4 = p Binomial Distribution ( 29 ( 29 ( 29 n x p p x n x X P x p x n x ,..., 1 , , 1 =- = = =- ( 29 p n b X , ~ ( 29 ( 29 ( 29 =-- =- + = n x x n x n p p x n p p 1 1 1 Binomial Theorem 1 ( 29 n p- 1 ( 29 1 1 1-- n p p n ( 29 2 2 1 2-- n p p n n p X = 0 X = 1 X = 2 X = n ( 29 ( 29 t p p e t M n t X all for 1- + = ( 29 ( 29 ( 29 p np X Var np X E- = = 1 , Binomial Distribution Example: 50 Multiple choice questions A B C D E Correct: 2 marks Incorrect: -0.5 mark X = no. of correct answers by pure guess Y = score of the test by pure guess ( 29 2 . , 50 ~ b X ( 29 ( 29 ( 29 10 2 . 50 = = X E ( 29 ( 29 ( 29 ( 29 8 8 . 2 . 50 = = X Var ( 29 ( 29 ( 29 ( 29 02992 . 8 . 2 . 15 15 35 15 15 50 = = = = C p X P ( 29 ( 29 ( 29...
View Full Document

This note was uploaded on 02/01/2012 for the course STAT 1801 taught by Professor Mrchung during the Fall '10 term at HKU.

Page1 / 62

stat1801_4 - Uniform Distribution ( 29 <...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online