{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

stat2303_assignment01_ans

stat2303_assignment01_ans - THE UNIVERSITY OF HONG KONG...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT2303/2803 PROBABILITY MODELLING/STOCHASTIC MODELS ASSIGNMENT 1 SOLUTION 1. From (1), 𝑃𝑃 ( 𝐴𝐴𝐴𝐴 | 𝐶𝐶 ) = 𝑃𝑃 ( 𝐴𝐴 | 𝐶𝐶 ) 𝑃𝑃 ( 𝐴𝐴 | 𝐶𝐶 ) (1) 𝑃𝑃 ( 𝐴𝐴𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐶𝐶 ) = 𝑃𝑃 ( 𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐶𝐶 ) 𝑃𝑃 ( 𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐶𝐶 ) 𝑃𝑃 ( 𝐴𝐴 | 𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐶𝐶 ) = 𝑃𝑃 ( 𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐶𝐶 ) 𝑃𝑃 ( 𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐶𝐶 ) ⇔ 𝑃𝑃 ( 𝐴𝐴 | 𝐴𝐴𝐶𝐶 ) = 𝑃𝑃 ( 𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐶𝐶 ) = 𝑃𝑃 ( 𝐴𝐴 | 𝐶𝐶 ) (2) And, 𝑃𝑃 ( 𝐴𝐴𝐴𝐴 | 𝐶𝐶 ) = 𝑃𝑃 ( 𝐴𝐴 | 𝐶𝐶 ) 𝑃𝑃 ( 𝐴𝐴 | 𝐶𝐶 ) (1) 𝑃𝑃 ( 𝐴𝐴𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐶𝐶 ) = 𝑃𝑃 ( 𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐶𝐶 ) 𝑃𝑃 ( 𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐶𝐶 ) 𝑃𝑃 ( 𝐴𝐴 | 𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐶𝐶 ) = 𝑃𝑃 ( 𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐶𝐶 ) 𝑃𝑃 ( 𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐶𝐶 ) ⇔ 𝑃𝑃 ( 𝐴𝐴 | 𝐴𝐴𝐶𝐶 ) = 𝑃𝑃 ( 𝐴𝐴𝐶𝐶 ) 𝑃𝑃 ( 𝐶𝐶 ) = 𝑃𝑃 ( 𝐴𝐴 | 𝐶𝐶 ) (3) So, (1) (2) (3) 2. 𝑃𝑃 ( 𝑌𝑌 = 1) = � 𝑃𝑃 ( 𝑋𝑋 = 𝑥𝑥 , 𝑌𝑌 = 1) 𝑥𝑥 = 1 9 + 1 3 + 1 9 = 5 9 𝑃𝑃 ( 𝑌𝑌 = 2) = � 𝑃𝑃 ( 𝑋𝑋 = 𝑥𝑥 , 𝑌𝑌 = 2) 𝑥𝑥 = 1 9 + 0 + 1 18 = 3 18 𝑃𝑃 ( 𝑌𝑌 = 3) = � 𝑃𝑃 ( 𝑋𝑋 = 𝑥𝑥 , 𝑌𝑌 = 3) 𝑥𝑥 = 0 + 1 6 + 1 9 = 5 18 𝐸𝐸 ( 𝑋𝑋 | 𝑌𝑌 = 1) = � 𝑥𝑥 𝑃𝑃 ( 𝑋𝑋 = 𝑥𝑥 , 𝑌𝑌 = 1) 𝑃𝑃 ( 𝑌𝑌 = 1) 𝑥𝑥 = 9 5 1 1 9 + 2 1 3 + 3 1 9 �� = 2 𝐸𝐸 ( 𝑋𝑋 | 𝑌𝑌 = 2) = � 𝑥𝑥 𝑃𝑃 ( 𝑋𝑋 = 𝑥𝑥 , 𝑌𝑌 = 2) 𝑃𝑃 ( 𝑌𝑌 = 2) 𝑥𝑥 = 18 3 1 1 9 + 2(0) + 3 1 18 �� = 5 3 𝐸𝐸 ( 𝑋𝑋 | 𝑌𝑌 = 3) = � 𝑥𝑥 𝑃𝑃 ( 𝑋𝑋 = 𝑥𝑥 , 𝑌𝑌 = 3) 𝑃𝑃 ( 𝑌𝑌 = 3) 𝑥𝑥 = 18 5 1(0) + 2 1 6 + 3 1 9 �� = 12 5 3. 𝑃𝑃 ( 𝑌𝑌 = 2) = � � 𝑃𝑃 ( 𝑋𝑋 = 𝑥𝑥 , 𝑍𝑍 = 𝑧𝑧 , 𝑌𝑌 = 2) 𝑧𝑧 𝑥𝑥 = 1 16 + 1 4 = 5 16
Image of page 1

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

View Full Document Right Arrow Icon
𝑃𝑃 ( 𝑋𝑋 = 1| 𝑌𝑌 = 2) = 𝑃𝑃 ( 𝑋𝑋 = 1, 𝑌𝑌 = 2) 𝑃𝑃 ( 𝑌𝑌 = 2) = 1 16 + 0 5 16 = 1 5 𝑃𝑃 ( 𝑋𝑋 = 2| 𝑌𝑌 = 2) = 𝑃𝑃 ( 𝑋𝑋 = 2, 𝑌𝑌 = 2) 𝑃𝑃 ( 𝑌𝑌 = 2) = 0 + 1 4 5 16 = 4 5 𝐸𝐸 ( 𝑋𝑋 | 𝑌𝑌 = 2) = 1 1 5 + 2 4 5 = 9 5 𝑃𝑃 ( 𝑌𝑌 = 2, 𝑍𝑍 = 1) = � 𝑃𝑃 ( 𝑋𝑋 = 𝑥𝑥 , 𝑌𝑌 = 2, 𝑍𝑍 = 1) 𝑥𝑥 = 1 16 + 0 = 1 16 𝑃𝑃 ( 𝑋𝑋 = 1| 𝑌𝑌 = 2, 𝑍𝑍 = 1) = 1
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.