Probability and Statistics

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2018 Q.32)** **1. Governing Concept & Definition**: - **Topic**: Statistics & Standard Normal Distribution Symmetry - For a standard normal variable $Z \sim \mathcal{N}(0, 1)$, the distribution is symmetric about 0: $P(Z^2 > 1) = P(Z > 1) + P(Z < -1) = 2 \times P(Z > 1)$ **2. Detailed Step-by-Step Calculation**: - **Given**: - $P(0 \le Z \le 1) = 0.34$ - **Total probability for half-distribution ($Z \ge 0$)**: $P(Z \ge 0) = 0.50$ - **Tail probability $P(Z > 1)$**: $P(Z > 1) = P(Z \ge 0) - P(0 \le Z \le 1) = 0.50 - 0.34 = 0.16$ - **Two-tailed probability $P(Z^2 > 1)$**: $P(Z^2 > 1) = 2 \times P(Z > 1) = 2 \times 0.16 = 0.32$ **3. Final Answer**: - **Calculated Value**: **0.32** - **Official GATE Accepted Range**: **0.32 to 0.32**