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stat1801_3 - Example:SicBo Toss3fairdice X=gainbyinonegame...

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    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 . 0 3600 1100 - = - Average gain per game ( 29 ( 29 ( 29 ( 29 ( 29 3056 . 0 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 : don’t bet!
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    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  2 ( 29 ( 29 ( 29 ( 29 = X x x p x X E log log Expected value of log(  ) ( 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 ……
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    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 0 , 10 1 10 = - t e t f t ( 29 10 10 1 0 10 = = - dt te T E t
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    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 . 0 20 4 . 0 10 2 . 0 0 1 . 0 10 = + + + - = S E ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 8 2 . 0 10 6 . 0 8 2 . 0 6 = + + = B E ( 29 [ ] ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 89 3 . 0 11 4 . 0 1 2 . 0 9 1 . 0 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 . 0 2 6 . 0 0 2 . 0 2 8 2 2 2 2 = + + - = - B E ( 29 [ ] 2 2 % 6 . 1 8 = - B E Population Mean : Population Variance : ( 29 X E = μ ( 29 ( 29 [ ] 2 2 μ σ - = = X E X Var ( 29 2 % 89 = S Var ( 29 2 % 6 . 1 = B Var ( 29 2 2 μ - = X E Population Standard Deviation : ( 29 X Var = σ
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    Mean and Variance Discrete                pmf Continuous            pdf Mean Variance ( 29 x f ( 29 = dx x xf μ ( 29 ( 29 - = dx x f x 2 2 μ σ ( 29 x p ( 29 = x xp μ ( 29 ( 29 - = x p x 2 2 μ σ ( 29 ( 29 c X aE c aX E + = + ( 29 ( 29 X Var a c aX Var 2 = +
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    Mean and Variance Example:  X  = size of random loss (in $1000) ( 29 0 , 2 1 2 = - x e x f x Mean loss ( 29 2 2 1 0 2 = = = - dx xe X E x μ Volatility of loss ( 29 8 2 1 0 2 2 2 = = - dx e x X E x ( 29 ( 29 4 2 2 2 = - = = μ σ X E X Var 2 = σ
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    Mean and Variance Example:  X  = size of random loss (in $1000) ( 29 0 , 2 1 2 = - x e x f x ( 29 2 = X E ( 29 4 = X Var Amount covered by writer of contract    Y  = 0.8 X ( 29 ( 29 6 . 1 8 . 0 = = X E Y E ( 29 ( 29 ( 29 56 . 2 8 . 0
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