Question

Let \( X \) be a discrete random variable with the probability mass function

\[ P(X = n) = \begin{cases} \frac{-2c}{n}, & n = -1, -2 \\ d, & n = 0 \\ \frac{cn}{n}, & n = 1, 2 \\ 0, & \text{otherwise} \end{cases} \]

where \( c \) and \( d \) are positive real numbers. If \( P(|X| \leq 1) = \frac{3}{4} \), then \( E(X) \) equals

• A. \( \frac{1}{12} \)
• B. \( \frac{1}{6} \)
• C. \( \frac{1}{3} \)
• D. \( \frac{1}{2} \)

Hint:

For calculating the expected value of a discrete random variable, multiply each value of the random variable by its corresponding probability and sum them.

Answer 1

The Correct Option is A

Step 1: Define the probability mass function.
Given the PMF for the random variable \( X \), we have the probabilities for each value of \( X \). The total probability must sum to 1, so we can use the condition \( P(|X| \leq 1) = \frac{3}{4} \) to calculate the constants \( c \) and \( d \).
Step 2: Use the condition \( P(|X| \leq 1) = \frac{3}{4} \).
For \( P(|X| \leq 1) \), the values of \( X \) that satisfy this condition are \( X = -1, 0, 1 \). Therefore, we can write:

\[ P(X = -1) + P(X = 0) + P(X = 1) = \frac{3}{4} \]

Substitute the values from the PMF and solve for \( c \) and \( d \).
Step 3: Calculate the expected value \( E(X) \).
Once we have the values of \( c \) and \( d \), we calculate the expected value using the formula:

\[ E(X) = \sum_{n} n \cdot P(X = n) \]

Substitute the PMF values and solve for \( E(X) \). The result is \( E(X) = \frac{1}{3} \).
Step 4: Conclusion.
The correct answer is (C) \( \frac{1}{3} \).