Question 1

Since the distribution is uniform, we know that the possibilities of \(X\) is equal to \(k^n\) and

\[1 = k^n - (k - 1)^n\]

Since \(k^n\) is the total number of possibilities and \((k - 1)^n\) are the possibilities that aren’t equal to 1, if Y is to equal any number \((Y = j)\), we get

\[For 1 \le j \le k, m(j) = \frac{(k - j + 1)^n - (j - k)^n}{k^n}\] Answer adapted from: https://math.dartmouth.edu/archive/m20f10/public_html/HW5Solutions.pdf

Question 2

a

Using a geometric distribution:

b

Using an exponential distribution:

c

Using a binomial distribution:

d

Using a poisson distribution: