Question

Baskets and apples. There are 2 baskets with a total of n apples inside. Every now and then, Wilbur chooses an apple uniformly at random and moves it from its current basket to the other. We construct the process X.

a. Calculate the probability distribution of X.
b. Determine the expected value of X.
c. Find the variance of X.
d. None of the above.

Answer 1

The random variable X indicates the number of apples in one basket. The expected value of X is n/2, while the variance would depend on the probabilities of selection.

Step-by-step explanation:

For the scenario described, the random variable X indicates the number of apples in one of the baskets at any given time. The probability distribution of X can be determined using a binomial distribution if we define a success as choosing an apple from the basket which originally contained more apples.

The expected value of X would logically be n/2 since the baskets should average out to contain half of the total number of apples over time if the selection of each apple is truly random. Regarding the variance, it would depend on the actual probabilities of the random selections, which have not been given in the details.

If the basket selection is truly random, we might assume that each apple has an equal chance of being selected, and thus the variance would depend on those equal probabilities.