stat1801_3

stat1801_3 - Mathematical Expectation Example: Sic Bo ( ? ?...

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Unformatted text preview: Mathematical Expectation Example: Sic Bo ( ? ? ) Toss 3 fair dice Bet option 2 : win $k if there are k dice faced up as six, otherwise lose $1 Bet option 1 : win $24 if same number on three dice, otherwise lose $1 X = gain by in one game ( 29 36 1 24 = = X P ( 29 36 35 1 =- = X P In 3600 games, gain ( 29 1100 3500 1 100 24- = - + Average gain per game 3056 . 3600 1100- =- Average gain per game ( 29 ( 29 ( 29 ( 29 ( 29 3056 . 1 1 24 24 3600 3500 1 100 24- =- = - + = = - + X P X P E(gain by ) = -7.87% E(gain by ) = -30.56% Better strategy : Best strategy : dont bet! Mathematical Expectation ( 29 ( 29 ( 29 = X x x xp X E Expected value of random variable X ( 29 ( 29 ( 29 = X x x p x X E 2 2 Expected value of X 2 ( 29 ( 29 ( 29 ( 29 = X x x p x X E log log Expected value of log( X ) ( 29 5 5 = E ( 29 ( 29 ( 29 ( 29 X E X E X E X X X E 10 3 5 10 3 5 2 2- + =- + ( 29 7 . 3 7 . 3 = E ( 29 ( 29 2 2 X E X E ( 29 ( 29 X E X E log log Mathematical Expectation Continuous random variable X ( 29 ( 29 - = dx x xf X E ( 29 ( 29 - = 2 2 dx x f x X E ( 29 ( 29 ( 29 - = log log dx x f x X E ( 29 ( 29 ( 29 ( 29 - = dx x f x g X g E Example: waiting time T ( 29 , 10 1 10 =- t e t f t ( 29 10 10 1 10 = = - dt te T E t Mean and Variance Example: investment Return on Stock-10% 0% 10% 20% Probability 0.1 0.2 0.4 0.3 Return on Bonds 6% 8% 10% Probability 0.2 0.6 0.2 ( 29 % 9 = S E ( 29 % 8 = B E ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 9 3 . 20 4 . 10 2 . 1 . 10 = + + +- = S E ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 8 2 . 10 6 . 8 2 . 6 = + + = B E ( 29 [ ] ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 89 3 . 11 4 . 1 2 . 9 1 . 19 9 2 2 2 2 2 = + +- +- =- S E ( 29 [ ] 2 2 % 89 9 =- S E ( 29 [ ] ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 6 . 1 2 . 2 6 . 2 . 2 8 2 2 2 2 = + +- =- B E ( 29 [ ] 2 2 % 6 . 1 8 =- B E Population Mean : Population Variance : ( 29 X E = ( 29 ( 29 [ ] 2 2 - = = X E X Var ( 29 2 % 89 = S Var ( 29 2 % 6 . 1 = B Var ( 29 2 2 - = X E Population Standard Deviation : ( 29 X Var = Mean and Variance Discrete pmf Continuous pdf Mean Variance ( 29 x f ( 29 = dx x xf ( 29 ( 29 - = dx x f x 2 2 ( 29 x p ( 29 = x xp ( 29 ( 29 - = x p x 2 2 ( 29 ( 29 c X aE c aX E + = + ( 29 ( 29 X Var a c aX Var 2 = + Mean and Variance Example: X = size of random loss (in $1000) ( 29 , 2 1 2 =- x e x f x Mean loss ( 29 2 2 1 2 = = = - dx xe X E x Volatility of loss ( 29 8 2 1 2 2 2 = = - dx e x X E x ( 29 ( 29 4 2 2 2 =- = = X E X Var 2 = Mean and Variance Example: X = size of random loss (in $1000) ( 29...
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stat1801_3 - Mathematical Expectation Example: Sic Bo ( ? ?...

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