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Question 53
GATE BT 2021 • Q53
NAT • 2 Marks
Section 1: Engineering Mathematics
Probability and Statistics
Subtopic: Discrete & Continuous Probability Distributions
Question Statement:
If the values of two random variables (X, Y) are (121, 360), (242, 364)
and (363, 362), the value of correlation coefficient between X and Y
(rounded off to one decimal place) is _
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2021 Q.53)**
**1. Governing Concept & Definition**:
- **Topic**: Statistics & Pearson Correlation Coefficient ($r$)
- For paired samples $(X_i, Y_i)$:
$r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}}$
**2. Detailed Step-by-Step Calculation**:
- **Given Data Points**: $(121, 360)$, $(242, 364)$, $(363, 362)$
- **Means**:
$\bar{X} = \frac{121 + 242 + 363}{3} = \frac{726}{3} = 242$
$\bar{Y} = \frac{360 + 364 + 362}{3} = \frac{1086}{3} = 362$
- **Deviations**:
- $i=1$: $x_1 - \bar{x} = 121 - 242 = -121$; $y_1 - \bar{y} = 360 - 362 = -2$
- $i=2$: $x_2 - \bar{x} = 242 - 242 = 0$; $y_2 - \bar{y} = 364 - 362 = 2$
- $i=3$: $x_3 - \bar{x} = 363 - 242 = 121$; $y_3 - \bar{y} = 362 - 362 = 0$
- **Covariance Numerator**:
$\sum (x_i - \bar{x})(y_i - \bar{y}) = (-121)(-2) + (0)(2) + (121)(0) = 242$
- **Variances**:
$\sum (x_i - \bar{x})^2 = (-121)^2 + 0^2 + 121^2 = 2 \times 121^2$
$\sum (y_i - \bar{y})^2 = (-2)^2 + 2^2 + 0^2 = 4 + 4 = 8$
$\text{Denominator} = \sqrt{2 \times 121^2 \times 8} = \sqrt{16 \times 121^2} = 4 \times 121 = 484$
- **Correlation Coefficient**:
$r = \frac{242}{484} = 0.5$
**3. Final Answer**:
- **Calculated Value**: **0.5**
- **Official GATE Accepted Range**: **0.5 to 0.5**
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