Question Bank • QB_EM_ADV_074
MSQ • 2 Marks
Section 1: Engineering Mathematics
Probability and Statistics
Subtopic: Poisson, normal and binomial distributions
Question Statement:
Which of the following statements regarding the normal distribution $\mathcal{N}(\mu, \sigma^2)$ is/are TRUE?
Options:
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(A) The probability density curve is symmetric about the line $x = \mu$
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(B) The mean, median, and mode coincide at $x = \mu$
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(C) The points of inflection of the bell-shaped curve occur at $x = \mu \pm \sigma$
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(D) Approximately $68.27\%$ of the total area under the curve lies within the interval $[\mu - \sigma, \mu + \sigma]$
Step-by-Step Solution & Explanation
1. **Symmetry (Option A)**: $f(\mu + x) = f(\mu - x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-x^2 / (2\sigma^2)}$ (True).
2. **Central Tendency Alignment (Option B)**: By symmetry and unimodality, mean = median = mode = $\mu$ (True).
3. **Inflection Points (Option C)**: $f''(x) = 0 \implies \frac{(x - \mu)^2}{\sigma^2} = 1 \implies x = \mu \pm \sigma$ (True).
4. **Empirical Rule (Option D)**: $\int_{\mu-\sigma}^{\mu+\sigma} f(x)\,dx = \Phi(1) - \Phi(-1) = 0.8413 - 0.1587 = 0.6826$ (True).
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