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GATE 2024 Chemical Engineering, Q12 · Mathematics

The correct answer is B, (−3, 1/4).

Question

Consider the normal probability distribution function f(x) =[4/2π]⋅e−8(x+3)2.= [4/\sqrt{2π}]\cdot e^{−8(x+3)²}. If μμ and σσ are the mean and standard deviation of f(x) respectively, then the ordered pair (μ,σ)(μ, σ) is

  • A. (3, 1/4)
  • B. (−3, 1/4)
  • C. (3, 4)
  • D. (−3, 4)

Official answer: (−3, 1/4)

Why

GATE 2024 official key: (B). Matching f(x) to 1/(σ2π)⋅e−(x−μ)2/(2σ2):1/σ=4⇒σ=1/4,1/(σ\sqrt{2π})\cdot e^{−(x−μ)²/(2σ²)}: 1/σ = 4 \Rightarrow σ = 1/4, and 1/(2σ2)=81/(2σ²) = 8 is consistent with σ=σ = 1/4; (x−μ)2=(x+3)2⇒μ=−3.(x−μ)² = (x+3)² \Rightarrow μ = −3.

Explanation cross-checked for consistency. Final answer matches the official IIT answer key.

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