{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

stat1801_3

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

This preview shows pages 1–8. Sign up to view the full content.

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!

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

View Full Document
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 ……
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

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

View Full Document
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
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 = +

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

View Full Document
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
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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### 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