heinz nixdorf institute university of paderborn algorithms and complexity algorithms for radio...

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

Algorithms for Radio Networks

Exercise 12

Stefan Rührupsr@upb.de

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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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?

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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Exercise 24

1.) r ≤ 1 2.) 1 ≤ r ≤ √2

s

t

s

t

A1

A2

P[R ≤ r] = / = r2 / 4

P[R ≤ r] = (8 A1 + 4 A2) / 4

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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Exercise 24

s

t

A1

A2

P[R ≤ r] = (8 A1 + 4 A2) / 4

=

A1

r

1

√r2-1

A2

= /2 - 2cos = 1/r

A2 = r2 · /(2)

A1 = √r2-1

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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Exercise 24

• r ≤ 1: P[R ≤ r] = r2 / 4 P[R = r] = r / 2

• 1 ≤ r ≤ √2: P[R ≤ r] =

P[R = r] =

E[R] ≈ 0.765Numerical integration of

where fR is the probability density function (PDF),

which is piecewise defined by the two functions above

=: F1(r)

=: F2(r)

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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Exercise 24

• Distribution function (cumulative probability)

0 F1(r) F2(r) 1

r

P[R ≤ r]

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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Exercise 24

• Probability Density Function (PDF) for 0 ≤ r ≤ √2

r

P[R = r]

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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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=1

X=2

Y=1 1/4 1/4

Y=2 1/4 1/4

X/Y

X=1

X=2

Y=1

1 2

Y=2

1/2 1

HEINZ NIXDORF INSTITUTEUniversity 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 vmin or vmax, i.e. the speed is vmin with probability 1/2

and vmax 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

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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Exercise 26

• P[V = vmin] = 1/2 and P[V = vmax] = 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.

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

11

Exercise 26

• P[V = vmin] = 1/2 and P[V = vmax] = 1/2.

• The expected speed is

The expected speed is also given by

So, this is another example, where

(see Exercise 25)

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

12

Exercise 26

• Show that

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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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,

HEINZ NIXDORF INSTITUTEUniversity 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 Rührupsr@upb.deF2.313