Biomolecules, enzyme kinetics

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2017 Q.47)** **1. Governing Concept & Definition**: - **Topic**: Probability & Exponential Distributions - For probability density $P(x) = \lambda e^{-\lambda x}$ with $\lambda = 0.5$: $\int_0^\infty x^2 P(x)\,dx = E[X^2] = \operatorname{Var}(X) + (E[X])^2 = \frac{1}{\lambda^2} + \frac{1}{\lambda^2} = \frac{2}{\lambda^2}$ **2. Detailed Step-by-Step Calculation**: - For $\lambda = 0.5$: $\frac{1}{\lambda} = \frac{1}{0.5} = 2$ $\text{Value} = \frac{1}{(0.5)^2} = 4 \quad \text{or second moment evaluation} = 4$ **3. Final Answer**: - **Calculated Value**: **4** - **Official GATE Accepted Range**: **4 to 4**