Subtopic: Discrete & Continuous Probability Distributions
Question Statement:
The probability density function of a random variable X is
2
( )
2
x
p x
e
. The
probability
(1
2)
P
X
(rounded off to two decimal places) is __________.
Official Answer Key:
0.11 to 0.13
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2022 Q.31)**
**1. Governing Concept & Definition**:
- **Topic**: Probability & Continuous Exponential Distributions
- Given probability density function:
$p(x) = 2 e^{-2x} \quad \text{for } x \ge 0$
- Cumulative probability over an interval $[a, b]$ is:
$P(a \le X \le b) = \int_a^b p(x)\,dx = \left[ -e^{-2x} \right]_a^b = e^{-2a} - e^{-2b}$
**2. Detailed Step-by-Step Calculation**:
- **Interval**: $1 \le X \le 2$
- **Evaluation**:
$P(1 \le X \le 2) = e^{-2(1)} - e^{-2(2)} = e^{-2} - e^{-4}$
- $e^{-2} \approx 0.135335$
- $e^{-4} \approx 0.018316$
$P = 0.135335 - 0.018316 \approx 0.1170 \approx 0.12$
**3. Final Answer**:
- **Calculated Value**: **0.12**
- **Official GATE Accepted Range**: **0.11 to 0.13**
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