Question

1. Suppose that the hourly wage W has a gamma distribution. Suppose that the mean value E (W ) of this variable is 20 USD per hour and the standard deviation σ(W) is 5 USD.

Recall that for gamma distribution W ∼ Gamma(α, λ)
E(W) = α/λ, Var(W) = α/λ2

Suppose that in the setting of the two envelope problem the job offer is applied not to annual salaries, but to hourly wages. The open envelope has an offer of 18 USD per hour. Calculate the values of the distribution density f(w) for w = 9and w = 36 (using R or other software). What is the more likely value? Which of the envelopes should be chosen?

(Two Envelope Problem:Suppose you are looking for a job. You are at the end of a tedious interview process, but finally you are hired by a company, and you are ready to take the offer. The only issue remaining to be discussed is your salary. The boss, who is genuinely nice but rather eccentric, suggests that you play a game. He puts two envelopes on the table and informs you that each envelope contains your contract with the exact salary figure, but one number is two times greater than the other. You are allowed to choose one of the envelopes, open it and read the number. Then you have to make the decision: will you take the open envelope, or opt for the closed one (without being able to read the number it contains).)

Answer 1