2. Lectures prepared by: Elchanan Mossel Yelena Shvets Introduction to probability Stat 134 FAll 2005 Berkeley Follows Jim Pitman’s book: Probability Section 2.2

3. Binomial Distribution: Mean Recall: the Mean  of a binomial(n,p) distribution is given by:  = #Trials £ P(success) = n p The Mean = Mode (most likely value). The Mean = “Center of gravity” of the distribution.

4. Standard Deviation The Standard Deviation (SD) of a distribution, denoted  measures the spread of the distribution. Def: The Standard Deviation (SD)  of the binomial(n,p) distribution is given by:

5. Binomial Distribution Example 1: Let’s compare Bin(100,1/4) to Bin(100,1/2).  1 = 25,  2 = 50  1 = 4.33,  2 = 5 We expect Bin(100,1/4) to be less spread than Bin(100,1/2). Indeed, you are more likely to guess the value of Bin(100,1/4) distribution than of Bin(100,1/2) since: For p=1/2 : P(50) ¼ 0.079589237; For p=1/4: P(25) ¼ 0.092132;

6. Histograms for binomial(100, 1/4) & binomial(100, 1/2) ;

7. Binomial Distribution Example 2: Let’s compare Bin(50,1/2) to Bin(100,1/2).  1 = 25,  2 = 50  1 = 3.54,  2 = 5 We expect Bin(50,1/2) to be less spread than Bin(100,1/2). Indeed, you are more likely to guess the value of Bin(50,1/2) distribution than of Bin(100,1/2) since: For n=100: P(50) ¼ 0.079589237; For n=50 : P(25) ¼ 0.112556;

8. Histograms for binomial(50, 1/2) & binomial(100,1/2)

9. ,  and the normal curve We will see today that the Mean (  ) and the SD (  ) give a very good summary of the binom(n,p) distribution via The Normal Curve with parameters  and .

10. binomial(n,½); n=50,100,250,500  25,  3.54  50,  5  125,  7.91  250,  11.2

11. Normal Curve  x y

12. The Normal Distribution a xb

13. a xb

14. Standard Normal Standard Normal Density Function: Corresponds to  = 0 and  =1

15. Standard Normal For the normal ( ,  ) distribution: P(a,b)=  (  b-  ) -  (  a-  ); Standard Normal Cumulative Distribution Function:

16. Standard Normal

17. z-z

18. Standard Normal For the normal ( ,  ) distribution: P(a,b)=  (  b-  ) -  (  a-  ); In order to prove this, suffices to show: P(- 1,a) =  (  a-  ); Standard Normal Cumulative Distribution Function:

19. Change of variables

20. Properties of  :  (0)=1/2;  (-z)=1 -  (z);  (- 1 )=0;  ( 1 )=1.  does not have a closed form formula!

21. Normal Approximation of a binomial For n independent trials with success probability p: where:

22. binomial(n,½); n=50,100,250,500  25,  3.54  50,  5  125,  7.91  250,  11.2

23. Normal( ,  );  25,  3.54  50,  5  125,  7.91  250,  11.2

24. Normal Approximation of a binomial For n independent trials with success probability p: The 0.5 correction is called the “continuity correction” It is important when  is small or when a and b are close.

25. Normal Approximation Question: Find P(H>40) in 100 tosses.

26. Normal Approximation to bin(100,1/2).  ((40-50)/5) =1-  (-2) =  (2) ¼ 0.9772 Exact = 0.971556 What do we get With continuity correction ?

27. When does the Normal Approximation fail? bin(1,1/2)  =0.5,  =0.5 N(0.5,0.5)

28. When does the NA fail? bin(100,1/100)  =1,  ¼ 1 N(1,1)

29. Rule of Thumb Normal works better: The larger  is. The closer p is to ½.

30. Fluctuation in the number of successes. From the normal approximation it follows that: P(  -  to  +  successes in n trials) ¼ 68% P(  -2  to  +2  successes in n trials) ¼ 95% P(  -3  to  +3  successes in n trials) ¼ 99.7% P(  -4  to  +4  successes in n trials) ¼ 99.99%

31. Fluctuation in the proportion of successes. Typical size of fluctuation in the number of successes is: Typical size of fluctuation in the proportion of successes is:

32. Square Root Law Let n be a large number of independent trials with probability of success p on each. The number of successes will, with high probability, lie in an interval, centered on the mean np, with a width a moderate multiple of. The proportion of successes, will lie in a small interval centered on p, with the width a moderate multiple of 1/.

33. Law of large numbers Let n be a number of independent trials, with probability p of success on each. For each  > 0; P(|#successes/n – p|<  ) ! 1, as n ! 1

34. Confidence intervals

35. Conf Intervals

36. is called a 99.99% confidence interval.

37. binomial(n,½); n=50,100,250,500  25,  3.54  50,  5  125,  7.91  250,  11.2

38. Normal( ,  );  25,  3.54  50,  5  125,  7.91  250,  11.2

39. bin(50,1/2)  = 25,  = 3.535534 bin(50,1/2) + 25  50,  =3.535534 Normal Approximation bin(100,1/2)  50,  =5 (bin(50,1/2) + 25)* 5/3.535534  50,  = 5

40. Notre Dame de Rheims

41. Quote Bientôt je pus montrer quelques esquisses. Personne n'y comprit rien. Même ceux qui furent favorables à ma perception des vérités que je voulais ensuite graver dans le temple, me félicitèrent de les avoir découvertes au « microscope », quand je m'étais au contraire servi d'un télescope pour apercevoir des choses, très petites en effet, mais parce qu'elles étaient situées à une grande distance […]. Là où je cherchais les grandes lois, on m'appelait fouilleur de détails. (TR, p.346)