MA519_AUG09

MA519_AUG09 - QUALIFYING EXAMINATION August 2009 MA 519 (M....

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

View Full Document Right Arrow Icon
QUALIFYING EXAMINATION August 2009 MA 519 (M. D. Ward) 1. Consider n 2 random points, placed uniformly and independently, onto the edge of a circle with circumference 1. [An “arc” denotes a path on the circle.] 1a. (5 pts) Find the density of the length of the arc that connects the first point to the closest of the other points. 1b. (3 pts) Find the density of the straight line distance (i.e., not on the circle) between the two points described in part 1a . 1c. (5 pts) Consider the length of the largest arc that contains none of the n points in its interior. Prove that the length of this largest arc goes to 0 in probability as n → ∞ . 2. (5 pts) Consider independent random variables X and Y , with X uniform on (0 , 2) and with Y uniform on (0 , 3). Let M = max( X,Y ) and m = min( X,Y ). Find P ( m 2 > M ). 3. (5 pts) Ten students organize a tournament. Each student competes against each other student exactly once; all competitions are independent. In each competition, both students
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

MA519_AUG09 - QUALIFYING EXAMINATION August 2009 MA 519 (M....

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online