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# Lectures prepared by: Elchanan Mossel Yelena Shvets

Lectures prepared by: Elchanan Mossel Yelena Shvets. Binomial Distribution. Toss a coin 100, what’s the chance of 60 ?. hidden assumptions: n-independence; probabilities are fixed. ={ }. Magic Hat:.

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## Lectures prepared by: Elchanan Mossel Yelena Shvets

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1. Lectures prepared by:Elchanan MosselYelena Shvets

2. Binomial Distribution • Toss a coin 100, what’s the chance of 60 ? hidden assumptions: n-independence; probabilities are fixed.

3. ={ } Magic Hat: Each time we pull an item out of the hat it magically reappears. What’s the chance of drawing 3 I-pods in 10 trials?

4. Suppose we roll a die 4 times. What’s the chance of k ? Let . ={} Binomial Distribution k=0

5. Suppose we roll a die 4 times. What’s the chance of k ? Let . ={} Binomial Distribution k=1

6. Suppose we roll a die 4 times. What’s the chance of k ? Let . ={} Binomial Distribution k=2

7. Suppose we roll a die 4 times. What’s the chance of k ? Let . ={} Binomial Distribution k=3

8. Suppose we roll a die 4 times. What’s the chance of k ? Let . ={} Binomial Distribution k=4

9. Suppose we roll a die 4 times. What’s the chance of k ? Let . ={} Binomial Distribution k=0 k=1 k=2 k=3 k=4

10. Binomial Distribution How do we count the number of sequences of length 4 with 3 . and 1 .? .1 1. 1.. ..1 .2. ...1 ..3. .3.. 1... ....1 ...4. ..6.. .4... 1....

11. Binomial Distribution This is the Pascal’s triangle, which gives, as you may recall the binomial coefficients.

12. 1a 1b (a + b) = 1b2 2ab 1a2 (a + b)2 = 1a3 (a + b)3 = 3a2b 3ab2 1b3 1a4 6a2b2 4ab3 1b4 4a3b (a + b)4 = Binomial Distribution

13. Newton’s Binomial Theorem

14. Binomial Distribution For n independent trials each with probability p of success and (1-p) of failure we have This defines the binomial(n,p)distribution over the set of n+1 integers {0,1,…,n}.

15. Binomial Distribution This represents the chance that in n draws there was some number of successes between zero and n.

16. Example: A pair of coins • A pair of coins will be tossed 5 times. • Find the probability of getting • on k of the tosses, k=0 to 5. binomial(5,1/4)

17. # =k Example: A pair of coins • A pair of coins will be tossed 5 times. • Find the probability of getting • on k of the tosses, k=0 to 5. To fill out the distribution table we could compute 6 quantities for k = 0,1,…5 separately, or use a trick.

18. Binomial Distribution:consecutive odds ratio Consecutive odds ratio relates P(k) and P(k-1).

19. Use consecutive odds ratio to quickly fill out the distribution table Example: A pair of coins for binomial(5,1/4): • k • 0 • 1 • 2 • 3 • 4 • 5

20. Binomial Distribution We can use this table to find the following conditional probability: P(at least 3 | at least 1 in first 2 tosses) = P(3 or more & 1 or 2 in first 2 tosses) P(1 or 2 in first 2)

21. bin(5,1/4): • k • 0 • 1 • 2 • 3 • 4 • 5 bin(2,1/4): • 2 • k • 0 • 1

22. Stirling’s formula How useful is the binomial formula? Try using your calculators to compute P(500 H in 1000 coin tosses) directly: My calculator gave me en error when I tried computing 1000!. This number is just too big to be stored.

23. The following is calledthe Stirling’s approximation. It is not very useful if applied directly : nn is a very big number if n is 1000.

24. Stirling’s formula:

25. Binomial Distribution Toss coin 1000 times; P(500 in 1000|250 in first 500)= P(500 in 1000 & 250 in first 500)/P(250 in first 500)= P(250 in first 500 & 250 in second 500)/P(250 in first 500)= P(250 in first 500) P(250 in second 500)/ P(250 in first 500)= P(250 in second 500)=

26. Binomial Distribution: Mean Question: For a fair coin with p = ½, what do we expect in 100 tosses?

27. Binomial Distribution:Mean • Recall the frequency interpretation: • p ¼ #H/#Trials • So we expect about 50 H !

28. Binomial Distribution:Mean • Expectedvalue or Mean(m) • of a binomial(n,p) distribution m = #Trials £ P(success) = n pslip !!

29. Binomial Distribution: Mode Question: What is the most likely number of successes? Mean seems a good guess.

30. Binomial Distribution:Mode Recall that To see whether this is the most likely number of successes we need to compare this to P(k in 100) for every other k.

31. Binomial Distribution:Mode • The most likely number of successes is called the mode of a binomial distribution.

32. Binomial Distribution:Mode If we can show that for some m P(1) · … P(m-1) · P(m) ¸ P(m+1) … ¸ P(n), then m would be the mode.

33. Binomial Distribution:Mode so we can use successive odds ratio: to determine the mode.

34. > > > > > > Binomial Distribution:Mode successive odds ratio: If we replace · with > the implications will still hold. So for m=bnp+pc we get that P(m-1) · P(m) > P(m+1).

35. mode = 50 Binomial Distribution:Mode - graph SOR k

36. Binomial Distribution:Mode - graph distr k mode = 50

37. Lectures prepared by:Elchanan MosselYelena Shvets

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