{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Example Class 4 (sln)

# Example Class 4 (sln) - The University of Hong Kong...

This preview shows pages 1–2. Sign up to view the full content.

The University of Hong Kong Department of Statistics and Actuarial Science STAT1801 Probability and Statistics: Foundations of Actuarial Science (10-11) Example Class 4 1. Let the pdf of 𝑋𝑋 be 𝑓𝑓 ( 𝑥𝑥 ) = 2 𝑥𝑥 0 𝑓𝑓𝑓𝑓𝑓𝑓 0 < 𝑥𝑥 < 1 𝑓𝑓𝑜𝑜ℎ𝑒𝑒𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 a. Find 𝐸𝐸 ( 𝑋𝑋 ) and 𝑉𝑉𝑉𝑉𝑓𝑓 ( 𝑋𝑋 ) . 𝐸𝐸 ( 𝑋𝑋 ) = � 𝑥𝑥 (2 𝑥𝑥 ) 1 0 𝑑𝑑𝑥𝑥 = 2 3 𝐸𝐸 ( 𝑋𝑋 2 ) = � 𝑥𝑥 2 (2 𝑥𝑥 ) 1 0 𝑑𝑑𝑥𝑥 = 1 2 𝑉𝑉𝑉𝑉𝑓𝑓 ( 𝑋𝑋 ) = 𝐸𝐸 ( 𝑋𝑋 2 ) [ 𝐸𝐸 ( 𝑋𝑋 )] 2 = 1 18 b. Find the k th moment 𝜇𝜇 𝑘𝑘 of 𝑋𝑋 for 𝑘𝑘 = 1,2, … 𝜇𝜇 𝑘𝑘 = 𝐸𝐸�𝑋𝑋 𝑘𝑘 = � 𝑥𝑥 𝑘𝑘 (2 𝑥𝑥 ) 1 0 𝑑𝑑𝑥𝑥 = 2 𝑘𝑘 + 2 2. In a game, three fair dice are rolled. You bet on one of the numbers; say you bet \$1 on number 6. If no 6 shows, you lose your money. If any 6’s show, you get the \$1 back, plus an extra dollar for each 6 that shows. That means, you get a total of \$2 if one 6 shows. Let

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 ]}

### Page1 / 3

Example Class 4 (sln) - The University of Hong Kong...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online