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# HW#8 - ChBE 521 Homework 8 due Monday November 30 1...

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ChBE 521 Homework 8 due Monday, November 30 1. Consider a game involving two six-sided dice. Plot the probability of observing each total value (with 2,3,...,12 on the horizontal axis) if the dice are fair. What is the probability that the 5th roll is the first appearance of a 4? What is the probability that at most 10 rolls are required for a 2 (”snake eyes”) to be observed? How many times, on average, must one roll the dice before a 7 is observed? 2. The central limit theorem of statistics states that the sum of many in- dependent random variables tends toward a normal distribution without regard to the shape of the underlying distribution. Consider the statis- tic X that is a sum of N random variables u j , j 1 , N , each uniformly distributed on [0 , 1], X = N X j =1 u j . For N = 100 , 1000 , 10000 plot in MATLAB the distribution of X as a histogram and fit it to a normal distribution pdf. As N increases, does P ( X ) indeed approach a normal distribution? Repeat the calculations for Y (with N = 100 , 1000 , 10000) defined as Y = N X j =1 w j u j , where the

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