Theoretical DistributionsMCQMTP May 19Question 3571 of 230
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What is the first quartile of X having the following probability density function? f(t)=142πe−(t−10)232\displaystyle f(t) = \frac{1}{4\sqrt{2\pi}} e^{-\frac{(t-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 of a Normal Distribution** We are given the probability density function (PDF): f(t)=142πe−(t−10)232f(t) = \frac{1}{4\sqrt{2\pi}} e^{-\frac{(t-10)^2}{32}} **Step 1: Compare with the standard normal PDF** The PDF of a normal distribution with mean μ\displaystyle \mu and standard deviation σ\displaystyle \sigma is: f(t)=1σ2πe−(t−μ)22σ2f(t) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(t-\mu)^2}{2\sigma^2}} Comparing the parameters: - Mean μ=10\displaystyle \mu = 10 - Exponent denominator: 2σ2=32  ⟹  σ2=16  ⟹  Standard Deviation σ=4\displaystyle 2\sigma^2 = 32 \implies \sigma^2 = 16 \implies \text{Standard Deviation } \sigma = 4 **Step 2: Calculate the theoretical First Quartile (Q1\displaystyle Q_1)** The first quartile Q1\displaystyle Q_1 of a normal distribution is given by: Q1=μ−0.6745σQ_1 = \mu - 0.6745\sigma Substitute 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 the options** - Mathematically, Q1=7.30\displaystyle Q_1 = 7.30. - If standard deviation is σ=6\displaystyle \sigma = 6 instead of 4\displaystyle 4: 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. - This indicates a typo in the PDF parameters in the textbook where σ=6\displaystyle \sigma=6 was used for calculations. 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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