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SAMPLING – EVALUATING ALGORITMS October 3: Lecture 9 Some simple Combinatorics • In how many ways can we order n objects? • How do we prove this? – …… induction anyone? • This is also called the number of permutations Some simple Combinatorics • In how many ways we can select k objects out of n? 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 • The number is • How do we prove this? Some simple Combinatorics • In how many ways we can select 2 objects out of 4? • Number the objects and count distinct pairs • Take all permutations and count the first pair 1 2 3 4 1 2 4 3 2 1 3 4 2 1 4 3 • Each pair is `overcounted’, by a factor of Binomial Distribution Bernoulli Trial • A rigged coin: Head Probability p, Tail probability 1-p • Suppose I throw the coin n times. What is the probability of getting k heads? 1 2 3 4 5 • What is the probability that the k red are heads and the n-k blue are tails ? • Going back to the original question (consider all k subsets): The Bernoullis Bernoulli From Wikipedia, the free encyclopedia Bernoulli can refer to: any one or more of the Bernoulli family of Swiss mathematicians in the 18th century, including: Daniel Bernoulli (1700–1782), developer of Bernoulli's principle Jacob Bernoulli (1654–1705), also known as Jean or Jacques, after whom Bernoulli numbers are named Johann Bernoulli (1667–1748) Johann III Bernoulli (1744–1807), also known as Jean, astronomers Nicolaus I Bernoulli (1687–1759) Nicolaus II Bernoulli (1695–1726) Binomial Distribution by picture Expected Value also called mean • A dice, a random variable X (can be 1…6) • What is its expected value? • A rigged coin, Xi=1 with probability p, X=0 with 1-p • In general we sum the products of probabilities * values Binomial! Variance measuring the wildness • Ok, a random variable has an expected value. But it can be always near it, or it can be very wild all over the place! • This is captured by variance. • We define another random variable:: • We define • Standard deviation • Something helpful: • So.. Var(X) = E[X2] – (E[X])2 Variance for Binomial • A rigged coin, Xi=1 with probability p, X=0 with 1-p • The variance of Xi Proof? The normal distribution a continuous probability • Probability density function • The probability makes sense only for intervals Sampling before polls • For simplicity say that everyone is going to vote for either democrats or republicans. Also assume that every voter in the USA can be reached by phone. • Problem: Estimate the influence of the two parties. • Solution: Take 1000 persons at random and ask them. • Say 550 say democrats, 450 say republicans. What can we infer? Sampling before polls confidence intervals • Percentage of democrats is with probability 95% in the interval • Percentage of democrats is with probability 97.5% at least Confidence Intervals How do we prove them? • We have a Binomial Distribution. • For very large n it looks like a normal distribution • We can then find the shaded area around the mean, by calculating an interval • Concretely we ask what is the interval that covers, say 95% of the total area? Sampling before polls • For simplicity say that everyone is going to vote for either democrats or republicans. Also assume that every voter in the USA can be reached by phone. • However, not everyone has a listed phone number. More females hide their numbers from the public. • We can do the same experiment as before. But now we have a biased estimator. The estimator random variable does not have the same mean as the actual random variable. • This is called BIAS. Confidence intervals in general In general we desire unbiased estimators with small variance