Let \(Z\) be a standard normal. Compute \(E[Z^{101}]\).
Explain why the normal approximation to the binomial doesn’t really make sense, no matter how big \(n\) is.
An outbreak is suspected on a large cruise ship with 4000 passengers. Each passenger has an approximately 1% chance of being infected, and the infections are independent. (Highly unrealistic!)
Estimate the probability that 46 or more people are infected by rounding the standard deviation to the nearest integer, and pretending that that’s good enough. Hint. \(\Phi(1) \approx 0.841\).
Explain why your answer to (a) is an underestimate.
Repeat (a), but now use the fact that 46 is approximately 95% of a standard deviation away from the mean, and \(\Phi(0.95) \approx 0.83\).
Let \(Z\) be a standard normal. Compute \(E[Z^{100}]\).