Theoretical DistributionsMCQMTP Oct 21 MTP Sep 24 IIQuestion 3585 of 230
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What is the first quartile of X having the following probability density function? F(x)=12πe−(x−10)232\displaystyle F(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{(x-10)^2}{32}} for −∞<x<∞\displaystyle -\infty < x < \infty

Options

A4
B5
C5.95
D6.75
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Correct Answer

✅ Option c — 5.95

All Options:

  • A4
  • B5
  • C5.95
  • D6.75

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Detailed Solution & Explanation

**First Quartile from PDF** The given probability density function is: f(x)=12πe−(x−10)232f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{(x-10)^2}{32}} **Step 1: Compare with standard normal PDF** The standard PDF form of a normal distribution is: f(x)=1σ2πe−(x−μ)22σ2f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} By comparing parameters: - Mean μ=10\displaystyle \mu = 10 - Exponent denominator: 2σ2=32  ⟹  σ2=16  ⟹  σ=4\displaystyle 2\sigma^2 = 32 \implies \sigma^2 = 16 \implies \sigma = 4 *Note on constant factor:* The PDF expression has a minor typo where the leading factor is written as 12π\displaystyle \frac{1}{\sqrt{2\pi}} instead of 142π\displaystyle \frac{1}{4\sqrt{2\pi}}. **Step 2: Calculate the theoretical First Quartile (Q1\displaystyle Q_1)** The first quartile of a normal distribution is: Q1=μ−0.6745σQ_1 = \mu - 0.6745\sigma For μ=10\displaystyle \mu = 10 and σ=4\displaystyle \sigma = 4: Q1=10−0.6745×4=10−2.698=7.302Q_1 = 10 - 0.6745 \times 4 = 10 - 2.698 = 7.302 **Step 3: Analyze options** - If standard deviation is σ=6\displaystyle \sigma = 6 (a standard variation of this question): Q1=10−0.6745×6=10−4.047=5.953≈5.95Q_1 = 10 - 0.6745 \times 6 = 10 - 4.047 = 5.953 \approx 5.95 which matches Option C. - Option C is the correct answer according to the textbook's key under the assumption σ=6\displaystyle \sigma=6. Hence, **Option C** is the correct answer.

About This Chapter: Theoretical Distributions

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Binomial, Poisson, Normal Distribution

This chapter covers Binomial, Poisson, Normal Distribution and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

Key Concepts to Understand

Function

A relation from a set A (Domain) to a set B (Codomain) such that every element in A is associated with exactly one element in B. Written as f: A→B. Types include Linear, Quadratic, Polynomial, and Exponential functions.

Probability

A numerical measure of the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). Formula: P(A) = Number of favorable outcomes / Total number of equally likely outcomes.

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