The GATE Grind

GATE 2024 CH – Question 12

Engineering Mathematics · Probability and Statistics: Random variables, Poisson, Normal and Binomial distributions · 1 mark · Multiple choice

Consider the normal probability distribution function $f(x) = \frac{4}{\sqrt{2\pi}}e^{-8(x+3)^2}$
If $\mu$ and $\sigma$ are the mean and standard deviation of $f(x)$ respectively, then the ordered pair $(\mu, \sigma)$ is

  1. $(3, \frac{1}{4})$
  2. $(-3, \frac{1}{4})$
  3. $(3, 4)$
  4. $(-3, 4)$

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Correct answer: (B) $(-3, \frac{1}{4})$

Explanation

Compare with $\frac{1}{\sigma\sqrt{2\pi}}e^{-(x - \mu)^2/(2\sigma^2)}$. The exponent $-8(x + 3)^2$ gives $\mu = -3$ and $\frac{1}{2\sigma^2} = 8$, so $\sigma = \frac{1}{4}$. The prefactor $\frac{1}{\sigma\sqrt{2\pi}} = \frac{4}{\sqrt{2\pi}}$ agrees.