Subtopic: Discrete & Continuous Probability Distributions
Question Statement:
Consider a random variable X with mean 𝜇= 0.1 and variance 𝜎
ଶ= 0.2. A new
random variable 𝑌= 2𝑋+ 1 is defined. The variance of the random variable Y
(rounded off to one decimal place) is _________________.
Official Answer Key:
0.8 to 0.8
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2022 Q.55)**
**1. Governing Concept & Definition**:
- **Topic**: Statistics & Properties of Linear Transformations of Random Variables
- For a random variable $X$ with variance $\sigma_X^2$, the variance of a linear transformation $Y = aX + b$ is:
$\sigma_Y^2 = \operatorname{Var}(aX + b) = a^2 \operatorname{Var}(X) = a^2 \sigma_X^2$
**2. Detailed Step-by-Step Calculation**:
- **Given**:
- $\mu_X = 0.1$, $\sigma_X^2 = 0.2$
- Transformation: $Y = 2X + 1 \implies a = 2, b = 1$
- **Variance of $Y$**:
$\sigma_Y^2 = 2^2 \times \sigma_X^2 = 4 \times 0.2 = 0.8$
**3. Final Answer**:
- **Calculated Value**: **0.8**
- **Official GATE Accepted Range**: **0.8 to 0.8**
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