Question

A random variable \( X \) takes values \( 0, 1, 2, 3 \) with probabilities \( \frac{2a+1}{30}, \frac{8a-1}{30}, \frac{4a+1}{30}, b \) respectively, where \( a, b \in \mathbb{R} \). Let \( \mu \) and \( \sigma \) respectively be the mean and standard deviation of \( X \) such that \( \sigma^2 + \mu^2 = 2 \). Then \( \frac{a}{b} \) is equal to :

• A. 12
• B. 3
• C. 60
• D. 30

Hint:

The relation \( \sigma^2 + \mu^2 = \sum x_i^2 p_i \) often simplifies problems where both mean and variance are mentioned.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The sum of all probabilities must be 1. The variance formula is \( \sigma^2 = E(X^2) - [E(X)]^2 \), so \( \sigma^2 + \mu^2 = E(X^2) \).

Step 2: Detailed Explanation:
1. Sum of probabilities = 1:
\[ \frac{2a+1 + 8a-1 + 4a+1}{30} + b = 1 \implies \frac{14a+1}{30} + b = 1 \implies b = \frac{29 - 14a}{30} \]
2. Given \( \sigma^2 + \mu^2 = E(X^2) = 2 \):
\[ E(X^2) = 0^2 \cdot P(0) + 1^2 \cdot P(1) + 2^2 \cdot P(2) + 3^2 \cdot P(3) = 2 \]
\[ 0 + 1 \cdot \frac{8a-1}{30} + 4 \cdot \frac{4a+1}{30} + 9b = 2 \]
\[ \frac{8a - 1 + 16a + 4}{30} + 9(\frac{29 - 14a}{30}) = 2 \]
\[ 24a + 3 + 261 - 126a = 60 \implies -102a = -204 \implies a = 2 \]
3. Calculate \( b \):
\[ b = \frac{29 - 14(2)}{30} = \frac{1}{30} \]
4. Calculate \( a/b \):
\[ \frac{a}{b} = \frac{2}{1/30} = 60 \]

Step 3: Final Answer:
The ratio \( a/b \) is 60.