Statistics

Problem Statement for "Pillars"

Problem Statement

On a horizontal line, there are two vertical pillars. The distance between their bottoms is w. The height of the first pillar is an integer, chosen uniformly at random in the range 1 through x, inclusive. The height of the second pillar is an integer, chosen uniformly at random in the range 1 through y, inclusive. The tops of both pillars will be connected by a straight piece of rope.

You are given the ints w, x, and y. Compute and return the expected length of the rope.

Definition

Class:
Pillars
Method:
getExpectedLength
Parameters:
int, int, int
Returns:
double
Method signature:
double getExpectedLength(int w, int x, int y)
(be sure your method is public)

Notes

  • Your return value must have a relative or an absolute error of less than 1e-9.
  • In this task, the expected rope length can be computed as the average rope length over all possible cases.

Constraints

  • w will be between 1 and 1000, inclusive.
  • x will be between 1 and 100,000, inclusive.
  • y will be between 1 and 100,000, inclusive.

Examples

  1. 1

    1

    1

    Returns: 1.0

    The rope always has a length of 1.

  2. 1

    5

    1

    Returns: 2.387132965131785

    There are 5 possible (equiprobable) cases in which the length of the rope is 1, sqrt(2), sqrt(5), sqrt(10) and sqrt(17). The correct answer is the arithmetic average of these 5 numbers.

  3. 2

    3

    15

    Returns: 6.737191281760445

  4. 10

    15

    23

    Returns: 12.988608956320535

  5. 1000

    100000

    100000

    Returns: 33381.38304701605

  6. 1

    99175

    56445

    Returns: 32073.471757648073

    // precision is not too great, but seems ok

  7. 741

    98283

    97708

    Returns: 32694.75809065241

  8. 772

    97431

    92415

    Returns: 31758.202091392817

  9. 987

    98228

    96988

    Returns: 32589.00970552789

  10. 890

    92714

    96949

    Returns: 31712.49684962615

  11. 884

    99111

    94672

    Returns: 32402.59188274667

  12. 741

    98835

    90385

    Returns: 31806.143660389673

  13. 729

    92658

    92366

    Returns: 30866.59996029441

  14. 746

    98939

    97438

    Returns: 32765.82263936255

  15. 808

    93569

    91507

    Returns: 30895.89432876109

  16. 849

    90577

    92202

    Returns: 30511.110044647055

  17. 963

    98047

    96381

    Returns: 32459.944104889895

  18. 917

    92247

    95421

    Returns: 31356.225216130268

  19. 976

    99917

    90443

    Returns: 32072.809658547325

  20. 706

    97598

    97448

    Returns: 32533.942518992055

  21. 755

    93628

    99334

    Returns: 32299.044059147163

  22. 1000

    94941

    97711

    Returns: 32184.135418201913

  23. 764

    91934

    99803

    Returns: 32193.015738418748

  24. 801

    95554

    90102

    Returns: 31080.165714689017

  25. 788

    92938

    99183

    Returns: 32183.11427643082

  26. 867

    94321

    90666

    Returns: 30917.68049883712

  27. 717

    98053

    97904

    Returns: 32686.415243153057

  28. 998

    97740

    90882

    Returns: 31646.771878422198

  29. 905

    92995

    99484

    Returns: 32261.73462507481

  30. 810

    96526

    93249

    Returns: 31700.36466772312

  31. 907

    97517

    96124

    Returns: 32321.451176681683

  32. 785

    96134

    96755

    Returns: 32181.491324592185

  33. 960

    90202

    97048

    Returns: 31415.623263082023

  34. 819

    97532

    94947

    Returns: 32137.140460237355

  35. 806

    97015

    98420

    Returns: 32612.35196213513

  36. 880

    93996

    94931

    Returns: 31530.840430364417

  37. 1

    100000

    100000

    Returns: 33333.33344872877

  38. 1000

    1

    100000

    Returns: 50028.496625165004

  39. 1

    1

    100000

    Returns: 49999.50006935564

  40. 100

    100000

    100000

    Returns: 33334.04348856876

  41. 1000

    100000

    10000

    Returns: 45374.871961122386

  42. 1000

    50000

    50000

    Returns: 16749.035733819004

  43. 10

    100000

    100000

    Returns: 33333.342735219885

  44. 44

    100000

    100000

    Returns: 33333.48670500142

  45. 1000

    99999

    99998

    Returns: 33380.88362205062

  46. 1000

    99999

    99999

    Returns: 33381.05009484778

  47. 998

    88888

    99999

    Returns: 31941.31591403921

  48. 1000

    99999

    100000

    Returns: 33381.216574215534

  49. 1000

    90000

    90000

    Returns: 30052.226096239192

  50. 1000

    92673

    92673

    Returns: 30942.03323926141

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