Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Central Limit Theorem (Dr. Monticino) Assignment Sheet Read Chapter 18 Assignment #11 (Due April 13th) Chapter 18 Exercise Set A: 1,2,3,5 Exercise Set B: 1,6 Exercise Set C: 2,3,5 Review Exercises: 1,2,4,5 , 7(which is right), 8,10,11 Test 2 on Wednesday April 13th over Chapters 13 through 18 Overview Central Limit Theorem Intuition Sums Averages Examples Central Limit Theorem: Sums For a large number of random draws, with replacement, the distribution of the sum approximately follows the normal distribution Mean of the normal distribution is N* (expected value for one repetition) SD for the sum (SE) is N This holds even if the underlying population is not normally distributed Central Limit Theorem: Averages For a large number of random draws, with replacement, the distribution of the average = (sum)/N approximately follows the normal distribution The mean for this normal distribution is (expected value for one repetition) The SD for the average (SE) is N This holds even if the underlying population is not normally distributed Probability Histograms In a probability histograms, the area of the bar represents the chance of a value happening as a result of the random (chance) process For many processes, the associated probability histogram can be derived explicitly Draw two numbers from the box and sum or average them. For other processes, it may be difficult to explicitly determine the probability histogram In many of these cases, the CLT indicates that the normal distribution gives a good approximation of the actual probability histogram Draw 1000 numbers from the box and sum or average them. 0 0 1 1 0 0 1 Empirical Histograms Empirical histograms are histograms based on actually performing an experiment many times If the experiment is repeated many, many times, then the empirical histogram should be close to the (theoretical) probability histogram Example: Sums Flip a fair coin and count the number of heads – 10 flips 20% Percent 15% 10% 5% 0% 2.00 4.00 6.00 Number of Heads 8.00 Example: Sums Flip a fair coin and count the number of heads –50 flips 10.0% 7.5% Percent 5.0% 2.5% 0.0% 15.00 20.00 25.00 30.00 Number of Heads 35.00 Example: Sums Flip a fair coin and count the number of heads – 100 flips 8% 6% Percent 4% 2% 40.00 50.00 Number of Heads 60.00 Example: Averages Flip a fair coin and determine the proportion of times heads occurs – 10 flips 25% 20% Percent 15% 10% 5% 0.00 0.20 0.40 0.60 Proportion of Heads 0.80 Example: Averages Flip a fair coin and determine the proportion of times heads occurs – 50 flips 10.0% Percent 7.5% 5.0% 2.5% 0.0% 0.30 0.40 0.50 0.60 Proportion of Heads 0.70 Example: Averages Flip a fair coin and determine the proportion of times heads occurs – 100 flips 8% 6% Percent 4% 2% (Dr. Monticino) 0.30 0.40 0.50 0.60 Proportion of Heads