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**Unformatted text preview: **Poisson distribution. 6) Using moment-generating functions, show that as , the gamma distribution ( , ), properly standardized, tends to the standard normal distribution. 7) A soft-drink vending machine is set so that the amount of drink dispensed is a random variable with a mean of 200 milliliters, and a standard deviation 1 of 15 milliliters. What is the probability that the average amount dispensed in a random sample of size 36 is at least 204 milliliters? (Hint: use the central limit theorem.) 8) Consider the position of a particle following a random walk: each minute the particle moves north or south by 50 cm, with probability p = 1 / 2. Use the central limit theorem to estimate the probability that the position of the particle will be within 400 cm of the start after 1 hour. 2...

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