Subtopic: Discrete & Continuous Probability Distributions
Question Statement:
The distribution of marks scored by a large class in an exam can be represented as a normal
distribution with mean u and standard deviation o. In a follow-up exam in the same class, everyone
scored 5 marks more than their respective score in the earlier exam. For this follow-up exam, the
distribution of marks can be represented as a normal distribution with mean j2 and standard deviation
02. Which one of the following is correct?
Options:
(A) | = |2; 0 > 02
(B) u < M2; 0 > 02
(C) M > 112: 0 < 02
(D) | < |2: 0 = 02
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2017 Q.27)**
**1. Governing Concept & Question Classification**:
- **Section**: BT
- **Question Type**: **MCQ** (1 Mark)
- **Topic**: Biostatistics / Probability Distributions (Normal Distribution & Central Limit Theorem)
**2. Detailed Step-by-Step Analysis**:
- The performance (marks scored) of a very large class of students in an examination is governed by the summation of numerous independent, small random variables (e.g., student aptitude, preparation time, test question variations, mood, health).
- By the **Central Limit Theorem**, when a continuous quantitative variable is influenced by the additive combination of many independent random factors, its distribution asymptotically converges toward a **Normal (Gaussian) distribution**:
$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$
- The normal distribution is symmetric and bell-shaped, characterized entirely by its mean ($\mu$) and standard deviation ($\sigma$).
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: A Poisson distribution models discrete, rare count events per unit time or space (e.g., radioactive decays, bacterial colonies per plate).
- **Option (B) [INCORRECT]**: An exponential distribution models the waiting time between Poisson process events and is highly skewed.
- **Option (C) [INCORRECT]**: A binomial distribution models the number of successes in a fixed number of binary (yes/no) Bernoulli trials.
- **Option (D) [CORRECT]**: Marks scored by a large population of students are continuous and canonically represented by a Normal distribution.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **D**
- **Scoring Criteria**: +1.0 Mark (Negative marking: -0.33 marks).
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