1 / 40

Lectures prepared by: Elchanan Mossel Yelena Shvets

Lectures prepared by: Elchanan Mossel Yelena Shvets. Binomial Distribution: Mean. Recall: the Mean (m) of a binomial(n,p) distribution is given by:. m = #Trials £ P(success) = n p. The Mean = Mode (most likely value). The Mean = “Center of gravity” of the distribution.

leia
Download Presentation

Lectures prepared by: Elchanan Mossel Yelena Shvets

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lectures prepared by:Elchanan MosselYelena Shvets

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

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

  4. 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: Binomial Distribution For p=1/2 : P(50) ¼ 0.079589237; For p=1/4 : P(25) ¼ 0.092132;

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

  6. 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: Binomial Distribution For n=100 : P(50) ¼ 0.079589237; For n=50 : P(25) ¼ 0.112556;

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

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

  9. binomial(n,½); n=50,100,250,500 m=25, s=3.54 m=50, s=5 m=125, s=7.91 m=250, s=11.2

  10. Normal Curve y m - s m m + s x

  11. The Normal Distribution a b x

  12. The Normal Distribution a b x

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

  14. Standard Normal Standard Normal Cumulative Distribution Function: • For the normal (,) distribution: • P(a,b) = F((b-m)/s) - F((a-m)/s);

  15. Standard Normal

  16. Standard Normal -z z

  17. Standard Normal Standard Normal Cumulative Distribution Function: • For the normal (,) distribution: • P(a,b) = F((b-m)/s) - F((a-m)/s); • In order to prove this, suffices to show: • P(-1,a) = F((a-m)/s);

  18. Change of variables

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

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

  21. binomial(n,½); n=50,100,250,500 m=25, s=3.54 m=50, s=5 m=125, s=7.91 m=250, s=11.2

  22. Normal(m,s); m=25, s=3.54 m=50, s=5 m=125, s=7.91 m=250, s=11.2

  23. 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.

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

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

  26. When does the Normal Approximation fail? bin(1,1/2) m=0.5,s=0.5 N(0.5,0.5)

  27. When does the NA fail? bin(100,1/100) m =1, s¼ 1 N(1,1)

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

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

  30. 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:

  31. 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/ .

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

  33. Confidence intervals

  34. Conf Intervals

  35. Conf Intervals is called a 99.99% confidence interval.

  36. binomial(n,½); n=50,100,250,500 m=25, s=3.54 m=50, s=5 m=125, s=7.91 m=250, s=11.2

  37. Normal(m,s); m=25, s=3.54 m=50, s=5 m=125, s=7.91 m=250, s=11.2

  38. bin(50,1/2) + 25 m=50, s =3.535534 bin(50,1/2) m = 25, s = 3.535534 Normal Approximation m=50, s=5 (bin(50,1/2) + 25)* 5/3.535534 m=50, s = 5 bin(100,1/2) m=50, s=5

  39. Notre Dame de Rheims

  40. Quote Bientôt je pus montrer quelques esquisses. Personnen'y comprit rien. Même ceux qui furent favorables à ma perception des vérités que je voulais ensuite graverdans le temple, me félicitèrent de les avoirdécouvertes au « microscope », quand je m'étais aucontraire servi d'un télescope pour apercevoir deschoses, très petites en effet, mais parce qu'elles étaient situées à une grande distance […]. Là où jecherchais les grandes lois, on m'appelait fouilleur dedétails. (TR, p.346)

More Related