1) A diner owner claims that daily sales are independent and random, with a mean of $950 and standard deviation of $125.
a) Assuming this, approximate the probability that, in a month of 30 days, the mean daily sales exceeds $1000.
b) Suppose that in the next 30 days the mean daily sales does exceed $1000. What might you suspect about the owner's claim? Explain in a line or two.
2) Recall that the sum of independent Poisson random variables has a Poisson distribution with parameter the sum of the parameters. We can use this to approximate a Poisson distribution by a normal. Suppose lambda is a large positive value; let X ~ Poisson(lambda) and let X1,...,Xn be i.i.d. from a Poisson (lambda/n) distribution. Then X and X1 + ...+ Xn have the same distribution. Use the central limit theorem to determine the approximate distribution of X.
3) Suppose the number of accidents in a factory over the course of a year is modeled by a Poisson distribution with mean 16.3. Under this assumption, consider the probability there are at least 20 accidents in the next year.
a) Write the probability exactly as a sum; you do not need to calculate it.
b) Use the result from problem (2) to approximate this probability.