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stat1801_4

# stat1801_4 - UniformDistribution 1 f x = b a 0 0 x a F x =...

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Uniform Distribution ( 29 < < - = otherwise 0 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 0 ( 29 2 a b X E + = ( 29 3 2 2 2 a ab b X E + + = ( 29 ( 29 12 2 a b X Var - =

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Uniform Distribution Example: Drop a needle on the floor θ θ Angle between needle tip and north direction ( 29 π θ 2 , 0 ~ U ( 29 π π θ = + = 2 0 2 E ( 29 ( 29 3 12 0 2 2 2 π π θ = - = Var 8 1 4 2 2 1 4 2 2 4 = - = - = π π π π π π θ π F F P
Uniform Distribution Example: Buffon’s Needle Problem D D D L    (<  D ) ( 29 ? line a cross = P Drop

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Uniform Distribution Example: Buffon’s Needle Problem D ( 29 < = θ sin 2 line a cross L X P P θ X 2 , 0 ~ D U X ( 29 π θ , 0 ~ U D L dxd D L π θ π π θ 2 2 0 2 sin 0 = = ∫ ∫ pD L 2 = π Monte Carlo
Uniform Distribution ( 29 1 , 0 ~ U X 0 1 1 2 - = X Y -1 1 X Y 2 = 0 2 ( 29 1 , 1 ~ 1 2 - - = U X Y

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Uniform Distribution ( 29 6 , 2 ~ - U X -2 6 X Y - = 6 0 8 8 6 X Y - = 0 1 ( 29 1 , 0 ~ 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 , ~

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Binomial Distribution Value of  X 0 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 0 4 C 1 4 C 2 4 C 3 4 C 4 ( 29 ( 29 ( 29 ( 29 4 , 3 , 2 , 1 , 0 , 8 . 0 2 . 0 4 4 = = = = - x x x X P x p x x ( 29 4096 . 0 0 = p ( 29 4096 . 0 1 = p ( 29 1536 . 0 2 = p ( 29 0256 . 0 3 = p ( 29 0016 . 0 4 = p
Binomial Distribution ( 29 ( 29 ( 29 n x p p x n x X P x p x n x ,..., 1 , 0 , 1 = - = = = - ( 29 p n b X , ~ ( 29 ( 29 ( 29 = - - = - + = n x x n x n p p x n p p 0 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 ,

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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 . 0 , 50 ~ b X ( 29 ( 29 ( 29 10 2 . 0 50 = = X E ( 29 ( 29 ( 29 ( 29 8 8 . 0 2 . 0 50 = = X Var ( 29 ( 29 ( 29 ( 29 02992 . 0 8 . 0 2 . 0 15 15 35 15 15 50 = = = = C p X P ( 29 ( 29 ( 29 0607 . 0 9393 . 0 1 1 15 14 0 50 15 = - = - = = = = x x x p x p X P ( 29 ( 29 25 5 . 2 50 5 . 0 2 - = - × - + × = X X X Y ( 29 ( 29 ( 29 0 25 10 5 . 2 25 5 . 2 25 5 . 2 = - × = - = - = X E X E Y E ( 29 ( 29 ( 29 ( 29 50 8 25 . 6 5 . 2 25 5 . 2 2 = × = = - = X Var X Var Y Var ( 29 ( 29 ( 29 4426 . 0 9 0 25 5 . 2 0 = = < - = < X P X P Y P ( 29 ( 29 ( 29 0 26 40 25 5 . 2 40 = - = X P X P Y P
Geometric Distribution ( 29 p Geog X ~ X = number of trials until the first success is obtained F X  = 1 FS X  = 2 FFS X  = 3 ( 29 ( 29 p X P p - = = = 1 1 1 ( 29 ( 29 ( 29 p p X P p - = = = 1 2 2 ( 29 ( 29 ( 29 p p X P p 2 1 2 2 - = = = ( 29 ( 29 ,... 3 , 2 , 1 , 1 1 = - = - x p p x p x

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stat1801_4 - UniformDistribution 1 f x = b a 0 0 x a F x =...

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