1. HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Algorithms for Radio Networks Exercise 12 Stefan Rührup sr@upb.de

2. HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 2 Exercise 24 Assume that a start node s in the center (0, 0) of a square [−1, 1] × [−1, 1] of edge length 2 chooses uniformly at random a target node t in this square 1.What is the cumulative probability function P[R ≤ r] for the distance R = |s-t| 2 between the start node and target node if r ≤ 1? 2.What is the cumulative probability function P[R ≤ r] if 1 ≤ r ≤ √2? 3.Compute the corresponding probability density function and draw the graph of the function. 4.What is the expected value of R?

3. HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 3 Exercise 24 1.) r ≤ 12.) 1 ≤ r ≤ √2 s t s t A1A1 A2A2 P[R ≤ r] = / =  r 2 / 4 P[R ≤ r] = (8 A 1 + 4 A 2 ) / 4

4. HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 4 Exercise 24 s t A1A1 A2A2 P[R ≤ r] = (8 A 1 + 4 A 2 ) / 4 = A1A1 r 1 √r 2 -1 A2A2    =  /2 - 2  cos  = 1/r A 2 =  r 2 ·  /(2  ) A 1 = √r 2 -1

5. HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 5 Exercise 24 r ≤ 1: P[R ≤ r] =  r 2 / 4 P[R = r] =  r / 2 1 ≤ r ≤ √2: P[R ≤ r] = P[R = r] = E[R] ≈ 0.765 Numerical integration of where f R is the probability density function (PDF), which is piecewise defined by the two functions above =: F 1 (r) =: F 2 (r)

6. HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 6 Exercise 24 Distribution function (cumulative probability) 0 F 1 (r) F 2 (r) 1 r P[R ≤ r]

7. HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 7 Exercise 24 Probability Density Function (PDF) for 0 ≤ r ≤ √2 r P[R = r]

8. HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 8 Exercise 25 Find a counter-example that disproves for independent random variables X and Y. Chose X = {1,2} and Y={1,2} with P[X=1] = P[X=2] = 1/2 and P[Y=1] = P[Y=1] = 1/2 E[X] =  k · P[X=k] = 3/2 E[Y] =  k · P[Y=k] = 3/2 E[X/Y] =  k · P[X/Y=k] = 9/8 P[X/Y]X=1X=2 Y=11/4 Y=21/4 X/YX=1X=2 Y=112 Y=21/21

9. HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 9 Exercise 26 (additional exercise) An object moves with a constant speed for a fixed distance d. The speed V is chosen uniformly at random between either v min or v max, i.e. the speed is v min with probability 1/2 and v max with probability 1/2. –What is the average speed v? –What is the expected speed E[V]? –Show that v ≤ E[V] If the speed is constant, one needs a time of t = d/v to cover a fixed distance d. The average speed is given by

10. HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 10 Exercise 26 P[V = v min ] = 1/2 and P[V = v max ] = 1/2. The average speed is given by E[D] = d is fixed. But what is the expected time E[T]? If the speed is constant, one needs a time of t = d/v to cover a fixed distance d.

11. HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 11 Exercise 26 P[V = v min ] = 1/2 and P[V = v max ] = 1/2. The expected speed is The expected speed is also given by So, this is another example, where (see Exercise 25)

12. HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 12 Exercise 26 Show that

13. HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 13 Exercise 26 What is the minimum of the function x+1/x for x > 0? (this is a relative minimum, because the second derivative is greater than 0) x+1/x = 2 for x=1 Therefore,

14. HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Thanks for your attention! End of the lecture Mini-Exam No. 4 on Monday 13 Feb 2006, 2pm, FU.511 (Mozart) Good Luck! Stefan Rührup sr@upb.de F2.313