Question

Let \( X \) be a discrete random variable with the probability mass function
\[ p(x) = k(1 + |x|)^2, \quad x = -2, -1, 0, 1, 2, \] where \( k \) is a real constant. Then \( P(X = 0) \) equals

• A. \( \frac{2}{9} \)
• B. \( \frac{2}{27} \)
• C. \( \frac{1}{27} \)
• D. \( \frac{1}{81} \)

Hint:

For discrete random variables, ensure the probabilities sum to 1, and then use the normalized function to calculate individual probabilities.

Answer 1

The Correct Option is C

Step 1: Normalizing the probability mass function.
The total sum of the probabilities must be equal to 1. Therefore, we first find \( k \) by solving: \[ \sum_{x = -2}^{2} p(x) = 1 \] This gives us the value of \( k \).
Step 2: Finding \( P(X = 0) \).
After finding \( k \), we substitute \( x = 0 \) into the probability mass function: \[ p(0) = k(1 + |0|)^2 = k \] Thus, \( P(X = 0) = \frac{1}{27} \).