{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

oldfinal1

# oldfinal1 - UCSD ECE153 Prof Young-Han Kim Handout#35...

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

UCSD ECE153 Handout #35 Prof. Young-Han Kim Thursday, May 26, 2011 Final Examination (Fall 2008) 1. Order statistics. Let X 1 , X 2 , X 3 be independent and uniformly drawn from the interval [0 , 1]. Let Y 1 be the smallest of X 1 , X 2 , X 3 , let Y 2 be the median (second smallest) of X 1 , X 2 , X 3 , and let Y 3 be the largest of X 1 , X 2 , X 3 . For example, if X 1 = . 3 , X 2 = . 1 , X 3 = . 7, then Y 1 = . 1 , Y 2 = . 3 , Y 3 = . 7. The random variables Y 1 , Y 2 , Y 3 are called the order statistics of X 1 , X 2 , X 3 . (a) What is the probability P { X 1 X 2 X 3 } ? (b) Find the pdf of Y 1 . (c) Find the pdf of Y 3 . (d) (Difficult.) Find the pdf of Y 2 . (Hint: Y 2 y if and only if at least two among X 1 , X 2 , X 3 are y .) 2. Fair coins. We are given two coins: Coin 1 with bias (=probability of heads) 1 / 2 and Coin 2 with random bias P Unif[0 , 1]. We pick one at random and flip it three times independently. The value of the bias does not change during the sequence of tosses. Let X be the number of heads.

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 ]}