# Question: Two envelopes each containing a check are placed in front

Two envelopes, each containing a check, are placed in front of you. You are to choose one of the envelopes, open it, and see the amount of the check. At this point, either you can accept that amount or you can exchange it for the check in the unopened envelope. What should you do? Is it possible to devise a strategy that does better than just accepting the first envelope?

Let A and B, A < B, denote the (unknown) amounts of the checks, and note that the strategy that randomly selects an envelope and always accepts its check has an expected return of (A + B)/2. Consider the following strategy: Let F(·) be any strictly increasing (that is, continuous) distribution function. Choose an envelope randomly and open it. If the discovered check has the value x, then accept it with probability F(x) and exchange it with probability 1 − F(x).

(a) Show that if you employ the latter strategy, then your expected return is greater than (A + B)/2.

Condition on whether the first envelope has the value A or B.

Now consider the strategy that fixes a value x and then accepts the first check if its value is greater than x and exchanges it otherwise.

(b) Show that, for any x, the expected return under the x-strategy is always at least (A + B)/2 and that it is strictly larger than (A + B)/2 if x lies between A and B.

(c) Let X be a continuous random variable on the whole line, and consider the following strategy: Generate the value of X, and if X = x, then employ the x-strategy of part (b). Show that the expected return under this strategy is greater than (A + B)/2.

Let A and B, A < B, denote the (unknown) amounts of the checks, and note that the strategy that randomly selects an envelope and always accepts its check has an expected return of (A + B)/2. Consider the following strategy: Let F(·) be any strictly increasing (that is, continuous) distribution function. Choose an envelope randomly and open it. If the discovered check has the value x, then accept it with probability F(x) and exchange it with probability 1 − F(x).

(a) Show that if you employ the latter strategy, then your expected return is greater than (A + B)/2.

Condition on whether the first envelope has the value A or B.

Now consider the strategy that fixes a value x and then accepts the first check if its value is greater than x and exchanges it otherwise.

(b) Show that, for any x, the expected return under the x-strategy is always at least (A + B)/2 and that it is strictly larger than (A + B)/2 if x lies between A and B.

(c) Let X be a continuous random variable on the whole line, and consider the following strategy: Generate the value of X, and if X = x, then employ the x-strategy of part (b). Show that the expected return under this strategy is greater than (A + B)/2.

## Answer to relevant Questions

Successive weekly sales, in units of one thousand dollars, have a bivariate normal distribution with common mean 40, common standard deviation 6, and correlation .6. (a) Find the probability that the total of the next 2 ...A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice, and if tails, then one-half of the value that appears on the die. Determine her expected winnings. The Conditional Covariance Formula. The conditional covariance of X and Y, given Z, is defined by Cov(X, Y|Z) = E[(X − E[X|Z])(Y − E[Y|Z])|Z] (a) Show that Cov(X, Y|Z) = E[XY|Z] − E[X|Z]E[Y|Z] (b) Prove the conditional ...An urn initially contains b black and w white balls. At each stage, we add r black balls and then withdraw, at random, r balls from the b + w + r balls in the urn. Show that E[number of white balls after stage t] = (b + w / ...Student scores on exams given by a certain instructor have mean 74 and standard deviation 14. This instructor is about to give two exams, one to a class of size 25 and the other to a class of size 64. (a) Approximate the ...Post your question