Question

If the random variables X and Y follows discrete uniform over set $\{0,1,...,n\}$ and $\{1,2,...,n\}$ respectively then $Var(X) - Var(Y)$ equals to

• A. $\frac{2n+1}{12}$
• B. $\frac{1}{12}$
• C. $0$
• D. $-\frac{1}{12}$

Hint:

The variance of a discrete uniform distribution is translation-invariant. As long as the number of points in the sequence is the same and the spacing is the same (1 unit), the variance will be identical.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:

X is discrete uniform on {0, 1, ..., n} (n + 1 points) and Y is discrete uniform on {1, 2, ..., n} (n points). Find Var(X) - Var(Y).

Step 2: Detailed Explanation:

• For a discrete uniform variable on N consecutive integers, the variance is \( \frac{N^2 - 1}{12} \).

• For X, N = n + 1: \( Var(X) = \frac{(n+1)^2 - 1}{12} = \frac{n^2 + 2n}{12} \).

• For Y, N = n: \( Var(Y) = \frac{n^2 - 1}{12} \).

• \( Var(X) - Var(Y) = \frac{n^2 + 2n - n^2 + 1}{12} = \frac{2n + 1}{12} \).

Step 3: Final Answer:
The correct option is (A) \( \frac{2n+1}{12} \).