Statistical Analysis of a certain Random Walk and its associated Game By Shalosh B. Ekhad Suppose that you start at the 0, and at each round, walk on the discrete lin\ e as follows with probability, 1/3, move , 1, units to the left with probability, 1/3, move , 2, units to the left with probability, 1/3, move , 6, units to the right Since the expected progress in each move is positive, sooner or later you wo\ uld make it to, 1 but since life is finite, we will only do it for up to, 1000, rounds, You do that until you reached, for the first time , 1, or to its right and then it ends or already moved, 1000, rounds, at which case you give up The probability that you achieved your goal is, 1.000000000 conditioned on you finishing in <=, 1000, rounds the expected duration until you reach, 1, or beyond for the first time is 4.248813452 the variance of the duration is 43.14345780 hence the standard deviation is, 6.568367971 The scaled , 3, -th moment about the mean is, 5.373931032 The scaled , 4, -th moment about the mean is, 50.84370478 Now let's turn it into a game, where two 1D random walkers, follow the above\ mentioned random walk, and take turns, independent of each other The player to first reach the goal of , or above is declared the winner. conditioned on both players finishing in <= , 1000, rounds, the probability o\ f the first-to-move player to win that stupid game is 0.5965524945 -------------------------------------------- This ends this article that took, 24.383, seconds to generate. Statistical Analysis of a certain Random Walk and its associated Game By Shalosh B. Ekhad Suppose that you start at the 0, and at each round, walk on the discrete lin\ e as follows with probability, 1/4, move , 1, units to the left with probability, 1/4, move , 2, units to the left with probability, 1/4, move , 3, units to the left with probability, 1/4, move , 8, units to the right Since the expected progress in each move is positive, sooner or later you wo\ uld make it to, 1 but since life is finite, we will only do it for up to, 1000, rounds, You do that until you reached, for the first time , 1, or to its right and then it ends or already moved, 1000, rounds, at which case you give up The probability that you achieved your goal is, 0.9999977469 conditioned on you finishing in <=, 1000, rounds the expected duration until you reach, 1, or beyond for the first time is 10.26252443 the variance of the duration is 710.8655235 hence the standard deviation is, 26.66206150 The scaled , 3, -th moment about the mean is, 8.158988840 The scaled , 4, -th moment about the mean is, 110.5944219 Now let's turn it into a game, where two 1D random walkers, follow the above\ mentioned random walk, and take turns, independent of each other The player to first reach the goal of , or above is declared the winner. conditioned on both players finishing in <= , 1000, rounds, the probability o\ f the first-to-move player to win that stupid game is 0.5650380950 -------------------------------------------- This ends this article that took, 20.100, seconds to generate. Statistical Analysis of a certain Random Walk and its associated Game By Shalosh B. Ekhad Suppose that you start at the 0, and at each round, walk on the discrete lin\ e as follows with probability, 1/5, move , 1, units to the left with probability, 1/5, move , 2, units to the left with probability, 1/5, move , 3, units to the left with probability, 1/5, move , 4, units to the left with probability, 1/5, move , 15, units to the right Since the expected progress in each move is positive, sooner or later you wo\ uld make it to, 1 but since life is finite, we will only do it for up to, 1000, rounds, You do that until you reached, for the first time , 1, or to its right and then it ends or already moved, 1000, rounds, at which case you give up The probability that you achieved your goal is, 0.9999999670 conditioned on you finishing in <=, 1000, rounds the expected duration until you reach, 1, or beyond for the first time is 9.437185752 the variance of the duration is 402.4190880 hence the standard deviation is, 20.06038604 The scaled , 3, -th moment about the mean is, 6.932769458 The scaled , 4, -th moment about the mean is, 82.88484039 Now let's turn it into a game, where two 1D random walkers, follow the above\ mentioned random walk, and take turns, independent of each other The player to first reach the goal of , or above is declared the winner. conditioned on both players finishing in <= , 1000, rounds, the probability o\ f the first-to-move player to win that stupid game is 0.5522202728 -------------------------------------------- This ends this article that took, 69.419, seconds to generate. Statistical Analysis of a certain Random Walk and its associated Game By Shalosh B. Ekhad Suppose that you start at the 0, and at each round, walk on the discrete lin\ e as follows with probability, 1/6, move , 1, units to the left with probability, 1/6, move , 2, units to the left with probability, 1/6, move , 3, units to the left with probability, 1/6, move , 4, units to the left with probability, 1/6, move , 5, units to the left with probability, 1/6, move , 18, units to the right Since the expected progress in each move is positive, sooner or later you wo\ uld make it to, 1 but since life is finite, we will only do it for up to, 1000, rounds, You do that until you reached, for the first time , 1, or to its right and then it ends or already moved, 1000, rounds, at which case you give up The probability that you achieved your goal is, 0.9992923163 conditioned on you finishing in <=, 1000, rounds the expected duration until you reach, 1, or beyond for the first time is 20.33973246 the variance of the duration is 3664.015734 hence the standard deviation is, 60.53111377 The scaled , 3, -th moment about the mean is, 7.285871698 The scaled , 4, -th moment about the mean is, 73.14695195 Now let's turn it into a game, where two 1D random walkers, follow the above\ mentioned random walk, and take turns, independent of each other The player to first reach the goal of , or above is declared the winner. conditioned on both players finishing in <= , 1000, rounds, the probability o\ f the first-to-move player to win that stupid game is 0.5409069858 -------------------------------------------- This ends this article that took, 82.306, seconds to generate.