Questions
Chicken and Egg
Recall in the Chicken-Egg problem that we have a chicken that lays eggs, where the number of eggs N follows the distribution N∼Pois(λ). Given that the chicken lays N eggs, the number of eggs that hatch X follows the distribution X∣N∼Bin(N,p).
Find E(X)
Use Adam's Law!
Find Var(X)
Use Eve's Law!
Consumerism
When customers enter a particular store, each makes a purchase with probability p, independently. Given that a customer makes a purchase, the amount spent has mean μ (in dollars) and variance σ2.
Find the mean of how much a random customer spends (note that the customer may spend nothing).
Let X be the amount a random customer spends at the store, and let I be the indicator that a random customer makes a purchase. We then apply the Law of Total Expectation:
Find the variance of how much a random customer spends (note that the customer may spend nothing).
Let X be the amount a random customer spends at the store, and let I be the indicator that a random customer makes a purchase. We then apply the Law of Total Expectation:
Trapped Miners
A miner is trapped in a mine containing 3 doors. The first door leads to a tunnel that will take him to safety after 3 minutes. The other two doors lead to tunnels that will return him to the mine after 5 and 7 minutes of travel each. The miner is equally likely to choose any of the doors at any time.
What is the expected amount of time until he reaches safety?
Let W be the total amount of time taken until he reaches safety, and D1, D2, and D3 be the events of going through doors 1, 2, and 3 respectively.
Using LoTE, we have
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