Assume I choose $n$ random integers such that the last digit is uniformly distributed. What is the distribution of the last digit of the sum?

Question

Say that I sample $n$ random integers from some random variable $X$. The distribution has the last digit of the integer uniformly distributed. I then take the samples and add them

$$ Y = x_1+x_2+x_3 + ... + x_n $$

What is the distribution of the last digit of $Y$? I want to also say uniform, but I'm not sure

Answer 1

Yes, it's uniform. In fact, no matter what the value of $x_1+\cdots+x_{n-1}$ is, the ten possibilities for the last digit of $x_n$ all give different values for the last digit of $Y$. Since the last digit of $x_n$ is equally likely to be any of the ten possibilities, the same is true for $Y$. You don't need to know that all your $x_i$ are uniform in the last digit; as long as at least one of them is, the answer is uniform.

Answer 2

Yes, it is uniform: one good way to see this is to realize that the last digit is just the number $\mod{10}$. In order to prove your statement, just prove

If $X$ and $Y$ are uniform $\mod{10}$ and independent, then $X+Y$ is uniform $\mod{10}$ as well.

Then use induction to get your statement. If you need a hint on this step, consider conditioning on $X \mod{10}$ first.