Let X be a random day of the week, coded so that Monday is 1, Tuesday is 2, etc. (so X takes values 1, 2, ..., 7, with equal probabilities). Let Y be the next day after X (again represented as an integer between 1 and 7). Do X and Y have the same distribution

Yes. Have the same distribution.

Step-by-step explanation:

Given that X is a random day of the week, coded so that Monday is 1, Tuesday is 2, etc. (so X takes values 1, 2, ..., 7, with equal probabilities).

Y is the next day after X

So Y can be written as X+1

But since highest value is 7, we can write

Y = X+1 mod 7

i. e. whenever X=7, Y = 8 = 1 again.

Thus Y can take values as

Y 1 2 3 4 5 6 7

P (Y) = P (X+1) = 1/7

i. e. Y also has the same distribution as X