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Parishioners arrive at church on Sunday morning according to a Poisson process at rate 2/minute starting...

Parishioners arrive at church on Sunday morning according to a Poisson process at rate 2/minute starting at 9:00am until 11:00am. Each parishioner wears a hat with probability 1/3, independent of other parishioners, and brings an umbrella with probability 1/4, independent of whether she wears a hat and independent of other parishioners. The cloakroom has umbrella stands and basketsfor hats.

(a) What is the probability that the cloakroom has exactly 99 hats at 11:00am?

(b) How much space for hats must the cloakroom have to be approximately 98% sure that it won’t run out of space? (Use the central limit theorem).

(e) What is the expected amount of time that elapses starting at 9:00am until there are at least 3 items in the cloakroom?

Math   Statistics-And-Probability   
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