![Reviewing Example
Each instance of coupon
collector problem has to
pass through these
stages. Does it give you
some inspiration to
calculate E[X] ?
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9](https://image.slidesharecdn.com/lecture-13-cs648-131101031009-phpapp02/85/Lecture-13-cs648-9-320.jpg)
![Expected duration of a random experiment
Let X denote the random variable for the duration of a randomized
experiment.
To calculate E[X], the following approach is sometimes useful:
• Partition the experiment into stages carefully.
• Calculate expected duration of each stage.
• Using linearity of expectation, calculate E[X].
In the next class, we shall discuss more non-trivial randomized algorithms
which are analyzed using this method.
33](https://image.slidesharecdn.com/lecture-13-cs648-131101031009-phpapp02/85/Lecture-13-cs648-33-320.jpg)
Lecture 13-cs648
- 1. Randomized Algorithms CS648 Lecture 13 Expected duration of a randomized experiment Part I 1
- 4. Example 1 4 5 3 2 3 1 3 3 1 2 2 1 5 1 3 5 5 5 1 2 1 2 5 4 4 1 2 3 3 2 2 3 3 3 1 1 5 1 3 5 3 4 Done in 14 samplings Done in 12 samplings 3 2 2 5 4
- 5. Coupon Collector Problem ? No easy way !! 5
- 6. Coupon Collector Problem no coupon seen all coupons seen This transition is not sudden. In fact it is a gradual transition through various discrete stages. Can you see these discrete stages ? 6
- 7. Coupon Collector Problem no coupon seen 1 2 3 4 all coupons seen This transition is not sudden. In fact it is a gradual transition through various discrete stages. Can you see these discrete stages ? 7
- 8. Reviewing Example 1 4 5 3 2 0 5 3 1 3 3 1 2 2 1 5 1 3 5 3 4 8
- 9. Reviewing Example Each instance of coupon collector problem has to pass through these stages. Does it give you some inspiration to calculate E[X] ? 1 4 5 3 2 1 0 3 2 1 3 3 2 1 0 5 0 5 1 3 3 1 1 2 4 2 3 2 5 3 3 5 1 2 5 4 3 3 1 3 4 3 2 5 4 3 2 1 4 2 2 1 5 1 2 3 1 3 5 4 1 5 2 5 9
- 11. Reviewing Example 1 4 5 3 2 0 1 3 2 1 3 3 3 1 2 5 4 2 1 5 1 3 5 3 4 11
- 13. 13
- 14. 14
- 16. DISCRETE RANDOM WALK ON A LINE 16
- 17. Discrete Random Walk 0 1 2 3 4 5 6 7 8 … n n+1 • Particle starts from origin • In each second, particle moves 1 unit to the left or to the right with equal probability. • While at origin, the particle moves to 1 always. Question: What is the expected number of steps of the random walk to reach milestone n ? 17
- 18. An example 0 1 2 3 4 5 6 7 8 … I, and perhaps you too, could not notice the walk. So let us trace the walk slowly. 18
- 19. Formalism 19
- 20. Careful look at the example 0 1 2 3 4 5 6 7 8 … Can you break the walk 08 into stages ? Think carefully … 20
- 21. Careful look at the example 0 1 2 3 4 5 6 7 8 … Walk starting from 0 and terminating at 5 21
- 22. Careful look at the example 0 1 2 3 4 5 6 7 8 … Walk starting from 0 and terminating at 5 Walk starting from 5 and terminating at 8 22
- 23. 23
- 25. 25
- 26. Conditional Expectation Ω A useful tool to calculate expected value of a random variable 26
- 27. 0 1 … ? 27
- 28. 0 1 … 28
- 29. 0 1 … 3 29
- 30. 30
- 32. 32
- 33. Expected duration of a random experiment Let X denote the random variable for the duration of a randomized experiment. To calculate E[X], the following approach is sometimes useful: • Partition the experiment into stages carefully. • Calculate expected duration of each stage. • Using linearity of expectation, calculate E[X]. In the next class, we shall discuss more non-trivial randomized algorithms which are analyzed using this method. 33