lecture18.ppt - Probability Theory Counting in Terms of...

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Probability Theory: Counting in Terms of Proportions Great Theoretical Ideas In Computer Science Steven Rudich, Anupam Gupta CS 15-251 Spring 2004 Lecture 18 March 18, 2004 Carnegie Mellon University
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A Probability Distribution HEIGHT Proportio n of MALES
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The Descendants Of Adam Adam was X inches tall. He had two sons One was X+1 inches tall One was X-1 inches tall Each of his sons had two sons ….
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X X-1 X+1 1 1 X-2 X+2 X X-3 X+3 X-1 X+1 X-4 X+4 X-2 X+2 X 15 6 6 5 5 10 10 15 20
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1 X-1 X+1 1 1 X-2 X+2 X X-3 X+3 X-1 X+1 X-4 X+4 X-2 X+2 X 15 6 6 5 5 10 10 15 20
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1 1 1 1 1 X-2 X+2 X X-3 X+3 X-1 X+1 X-4 X+4 X-2 X+2 X 15 6 6 5 5 10 10 15 20
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1 1 1 1 1 1 1 2 X-3 X+3 X-1 X+1 X-4 X+4 X-2 X+2 X 15 6 6 5 5 10 10 15 20
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1 1 1 1 1 1 1 2 1 1 3 3 X-4 X+4 X-2 X+2 X 15 6 6 5 5 10 10 15 20
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1 1 1 1 1 1 1 2 1 1 3 3 1 1 4 4 6 15 6 6 5 5 10 10 15 20
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1 1 1 1 1 1 1 2 1 1 3 3 1 1 4 4 6 15 6 6 5 5 10 10 15 20 In n th generation, there will be 2 n males, each with one of n+1 different heights: h 0 < h 1 < . . .< h n . h i = (X – n + 2i) occurs with proportion
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Unbiased Binomial Distribution On n+1 Elements. Let S be any set {h 0 , h 1 , …, h n } where each element h i has an associated probability Any such distribution is called an Unbiased Binomial Distribution or an Unbiased Bernoulli Distribution .
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As the number of elements gets larger, the shape of the unbiased binomial distribution converges to a Normal (or Gaussian) distribution. Standard Deviation Mean
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1 1 4 4 6 Coin Flipping in Manhattan At each step, we flip a coin to decide which way to go. What is the probability of ending at the intersection of Avenue i and Street (n-i)
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1 1 4 4 6 Coin Flipping in Manhattan At each step, we flip a coin to decide which way to go. What is the probability of ending at the intersection of Avenue i and Street (n-i)
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Coin Flipping in Manhattan At each step, we flip a coin to decide which way to go. What is the probability of ending at the intersection of Avenue i and Street (n-i) 1 1 4 4 6
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Coin Flipping in Manhattan At each step, we flip a coin to decide which way to go. What is the probability of ending at the intersection of Avenue i and Street (n-i) 1 1 4 4 6
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Coin Flipping in Manhattan At each step, we flip a coin to decide which way to go. What is the probability of ending at the intersection of Avenue i and Street (n-i) 1 1 4 4 6
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Coin Flipping in Manhattan 2 n different paths to level n, each equally likely. The probability of i heads occurring on the path we generate is: 1 1 4 4 6
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n-step Random Walk on a line Start at the origin: at each point, flip an unbiased coin to decide whether to go right or left. The probability that, in n steps, we take i steps to the right and n-i to the left (so we are at position 2i-n) is: 1 1 4 4 6
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n-step Random Walk on a line Start at the origin: at each point, flip an unbiased coin to decide whether to go right or left.
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