Question

Let \( X \) be a discrete random variable uniformly distributed over \( \{-10, -9, \dots, 9, 10\} \). Which of the following random variables is/are uniformly distributed?

• A. \( X^2 \)
  (B) \( X^3 \)
  (C) \( (X-5)^2 \)
  (D) \( (X+10)^2 \)
• B.
• C.
• D.

Hint:

Uniform distribution requires all outcomes to have the same probability. Verify this by analyzing the range of values.

Answer 1

The Correct Option is B

Step 1: Checking uniformity. For \( X^2 \), values like \( 0, 1, 4, \dots, 100 \) are not uniformly distributed. For \( X^3 \), all values are unique, so it is uniformly distributed. For \( (X-5)^2 \), repeated values mean it is not uniformly distributed. For \( (X+10)^2 \), all values are unique, so it is uniformly distributed.