HUDM4122 Probability and Statistical Inference. March 4, 2015

Transcription

1 HUDM4122 Probability and Statistical Inference March 4, 2015

2 First things first

3 The Exam Due to Monday s class cancellation Today s lecture on the Normal Distribution will not be covered on the Midterm However, the previous lecture, on the Binomial Distribution, will be covered on the midterm

4 Apologies For the fact that this HW still included normal distribution problems We will go over those in the first class after the exam I was impressed with how well you all did despite the extra challenge

5 In the last class We studied the Binomial Probability Distribution The distribution of probability coming from a set of trials that have two possible outcomes

6 HW (Binomial)

7 P1 Your friend throws a ball at a target on Coney Island 100 times; how many times does it hit? Which of these is not a binomial experiment? You and your 6 friends buy a pizza with 10 slices, 4 pepperoni. If each friend takes one slice, at random, how many slices of pepperoni should you expect will be taken? You use an automated detector to figure out how many of 50 randomly sampled students gamed the system. You select 6 climate science professors out of all universities in the country. How many of them will say global warming is real?

8 P1 Which of these is not a binomial experiment? Your friend throws a ball at a target on Coney Island 100 times; how many times does it hit? You and your 6 friends buy a pizza with 10 slices, 4 pepperoni. If each friend takes one slice, at random, how many slices of pepperoni should you expect will be taken? You use an automated detector to figure out how many of 50 randomly sampled students gamed the system. You select 6 climate science professors out of all universities in the country. How many of them will say global warming is real?

9 P1: Common other answer (I can see arguments for it) Your friend throws a ball at a target on Coney Island 100 times; how many times does it hit? Which of these is not a binomial experiment? You and your 6 friends buy a pizza with 10 slices, 4 pepperoni. If each friend takes one slice, at random, how many slices of pepperoni should you expect will be taken? You use an automated detector to figure out how many of 50 randomly sampled students gamed the system. You select 6 climate science professors out of all universities in the country. How many of them will say global warming is real?

10 P2 For the following probability distribution, what is the mean? X P(X)

11 0(0.1)+1(0.3)+2(0)+3(0.3)+4(0.3) For the following probability distribution, what is the mean? X P(X)

12 For the following probability distribution, what is the mean? X P(X)

13 2.4 For the following probability distribution, what is the mean? X P(X)

14 Common Wrong Answer: ( )/5=0.2 For the following probability distribution, what is the mean? X P(X)

15 P9 For the following binomial probability distribution, what is the P(X=5)? N = 10 P = 0.5 Q = 0.5

16 For the following binomial probability distribution, what is the P(X=5)? N = 10 P = 0.5 Q = 0.5

17 For the following binomial probability distribution, what is the P(X=5)? N = 10 P = 0.5 Q = 0.5

18 In the last class We studied the Binomial Probability Distribution

19 Before we move on Any questions about binomial probability distributions?

20 Today Chapter 6 in Mendenhall, Beaver, & Beaver Normal Probability Distribution We ll start today We ll finish Wednesday

21 So far We ve talked about discrete probabilities, that can be written out in a table x P(x) 0 1/4 1 1/2 2 1/4

22 So far Though, of course, for 4000 trials, writing out that table would take an awfully long time

23 But what about Distributions of continuous (numerical) variables?

24 But what about Distributions of continuous (numerical) variables? These variables could have an infinite number of potential values

25 But what about Distributions of continuous (numerical) variables? These variables could have an infinite number of potential values So you can t write them in a table

26 So, whereas The histograms of discrete variables look blocky p(x) x

27 The histograms of continuous variables look smooth

28 Why? For a discrete variable, with overall mean 7, individual cases might be measured as 6 or 8 or even 5 Whereas for a continuous variable, with overall mean 6.421, individual cases might be measured as any number, with decreasing probability as you get further away from 6.421

29 For discrete distributions The probability of any value x is found to be P(x), a known probability for that value

30 For continuous distributions The probability of any value x is found by a formula f(x), which represents the density of cases likely to be found at value x More dense Less dense Even less dense

31 Because of this Probability distributions for continuous variables Are also called probability density functions

32 Commonality between Discrete and Continuous For Discrete Probabilities for all cases add to 1 For Continuous Probability under curve adds to 1

33 Similarity between Discrete and For Discrete Continuous Probability of value between a and b is sum of probability for all values between a and b For Continuous Probability of value between a and b is area under curve between a and b

34 How do we find the area under that curve? For some functions, it s not that bad

35 Flat Distribution Also called the uniform distribution

36 The Flat Distribution F(X)= 0 if X<i OR X>j F(X)=1 if X>i AND x<j Where i<j

37 Example (From Book) i=-0.5 J=+0.5 F(X)= 0 if X<i OR X>j F(X)=1 if X>i AND x<j Where i<j

38 Example (From Book) i=-0.5 J=+0.5 F(X)= 0 if X<-0.5 OR X>+0.5 F(X)=1 if X>-0.5 AND x<+0.5 Where -0.5<+0.5

39 Example (From Book) What is probability -0.2<x<+0.2? F(X)= 0 if X<-0.5 OR X>+0.5 F(X)=1 if X>-0.5 AND x<+0.5 Where -0.5<+0.5

40 Example (From Book) What is probability -0.2<x<+0.2? F(X)= 0 if X<-0.5 OR X>+0.5 F(X)=1 if X>-0.5 AND x<+0.5 Where -0.5<+0.5 So for -0.2<x<0.2, F(X)=1 Does everyone see why?

41 Example (From Book) What is probability -0.2<x<+0.2? F(X)= 0 if X<-0.5 OR X>+0.5 F(X)=1 if X>-0.5 AND x<+0.5 Where -0.5<+0.5 Rectangle width = -0.2 to 0.2 = 0.4 Rectangle height = 1

42 Example (From Book) What is probability -0.2<x<+0.2? F(X)= 0 if X<-0.5 OR X>+0.5 F(X)=1 if X>-0.5 AND x<+0.5 Where -0.5<+0.5 Rectangle width = -0.2 to 0.2 = 0.4 Rectangle height = 1 0.4*1=0.4, so P(-0.2<x<+0.2)=0.4

43 You Try It i=3 J=5 F(X)= 0 if X<i OR X>j F(X)=0.5 if X>i AND x<j Where i<j What is P(4.5<x<5)?

44 You Try It i=3 J=5 F(X)= 0 if X<i OR X>j F(X)=0.5 if X>i AND x<j Where i<j What is P(x>4.8)?

45 Questions? Comments?

46 BTW Since Probability of value between a and b is the area under curve between a and b The probability for any specific value is defined as 0 I.e. P(a) = 0 P(b) = 0

47 Which implies P(x>=a) = P(x>a) Not true for discrete random variables

48 Other functions are a bit trickier than the Flat Distribution

49 The one we ll focus on today is the Normal Distribution Also called the Gaussian distribution The version of the Normal Distribution that is almost always used is also called the Z distribution We ll discuss why in a bit

50 Why s it called the normal distribution? It s a good approximation for a lot of realworld data

51 Normal Distribution

52 µ = mean σ = standard deviation π = e =

53 How do we find the area under that curve? Option 1: Break out your integral calculus

54 How do we find the area under that curve? Option 2: Break out someone else s integral calculus

55 How do we find the area under that curve? Option 2: Break out someone else s integral calculus (Aka a table, or Microsoft Excel)

56 Normal Distribution (µ = 0, σ = 1)

57 Standardized Normal Distribution (µ = 0, σ = 1)

58 Standardized Normal Distribution P(X<0)=0.5 P(X>0)=0.5

59 Standardized Normal Distribution P(X<-1.96)=0.025

60 Standardized Normal Distribution P(X>1.96)=0.025

61 Standardized Normal Distribution P(-1.96>X>1.96)=0.95

62 Questions? Comments?

63 Remember z-scores?

64 Z-score formula z = z = The deviation, divided by the standard deviation

65 The Standardized Normal Distribution shows the probabilities of the values of Z That s why it s also called the Z distribution!

66 Standardized Normal Distribution (µ = 0, σ = 1) Z=0 Z=-2 Z=2

67 Recall from earlier in the semester If your data is normally distributed 68% of your data will be between -1 SD of the mean, and +1 SD of the mean z between -1 and +1 95% of your data will be between -2 SD of the mean, and +2 SD of the mean z between -2 and % of your data will be between -3 SD of the mean, and +3 SD of the mean z between -3 and +3

68 Ryan s Bad Advice For the Lovelorn If you tell someone they re 3 SD better than the mean, you re telling them they re better than 99.7%+1.5%=99.85% of other people

69 Comments? Questions?

70 Getting cumulative probabilities So, let s say you want to know the probability of Z<0 Once again, you can Do the integral calculus Look in a table Use Excel

71 Looking in a table Appendix I, Table 3 Page 664 in MBB

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74 What s the probability Z<-1.96?

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77 What s the probability Z<-1.64?

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80 What s the probability Z>-1.64?

81 What s the probability Z>-1.64? =0.9495

82 What s the probability -1.96<X<-1.64?

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85 What s the probability -1.96<X<-1.64? Between and =

86 What s the probability -1.96<X<+1.96?

87 What s the probability -1.96<X<+1.96? Between and = 0.95

88 What s the probability -1.96<X<+1.96? Between and = 0.95 In other words, 95% of the probability distribution is between and 1.96

89 Questions? Comments?

90 Doing it in Excel =NORMDIST(X,0,1,TRUE) For example =NORMDIST(-1.96,0,1,TRUE) Equals 0.025

91 Standardizing a Distribution Assuming data is normally distributed You can standardize that data, regardless of what its original mean and standard deviation were

92 Example Undergraduates rate their professor s quality The average rating is 3.8, SD is 0.4 What is the probability that a professor gets 4.5 or better?

93 Example Undergraduates rate their professor s quality The average rating is 3.8, SD is 0.4 What is the probability that a professor gets 4.5 or better? Z = (.. ). =.. =

94 Example Undergraduates rate their professor s quality The average rating is 3.8, SD is 0.4 What is the probability that a professor gets 4.5 or better? Z = (.. ). =.. = P(Z < 1.75) = 0.96

95 Example Undergraduates rate their professor s quality The average rating is 3.8, SD is 0.4 What is the probability that a professor gets 4.5 or better? Z = (.. ). =.. = P(Z < 1.75) = 0.96 thus P(Z>1.75) = 0.04

96 Example Z = (.. ). =.. = P(Z < 1.75) = 0.96 thus P(Z>1.75) = 0.04 So 4% of professors can be expected to get a rating of 4.5 or better

97 Try It In Solver-Explainer Pairs According to Stanford-Binet s totally discredited definition of a genius, a genius has an IQ of 140 or higher

98 Try It In Solver-Explainer Pairs According to Stanford-Binet s totally discredited definition of a genius, a genius has an IQ of 140 or higher Now we know it simply requires working at an Apple Store

99 Try It In Solver-Explainer Pairs According to Stanford-Binet s totally discredited definition of a genius, a genius has an IQ of 140 or higher The average IQ is 140, SD is 15 What is the probability that a person is a genius?

100 Questions? Comments?

101 Final questions or comments for the day?

102 Review sessions Thursday 10am and 1pm Location still TBD

103 3/9 Exam 1 Upcoming Classes