Theoretical DistributionsMCQMTP Dec 2022 Series IIQuestion 3540 of 230
All Questions

What is the mean of X having the following density function? f(x)=14π2e−(x−10)232\displaystyle f(x) = \frac{1}{\sqrt{4\pi}2} e^{-\frac{(x-10)^2}{32}} for −∞<x<∞\displaystyle -\infty < x < \infty

Options

A10\displaystyle 10
B4\displaystyle 4
C40\displaystyle 40
DNone of these
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Correct Answer

✅ Option a — 10\displaystyle 10

All Options:

  • A10\displaystyle 10
  • B4\displaystyle 4
  • C40\displaystyle 40
  • DNone of these

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Detailed Solution & Explanation

**Mean of a Normal Distribution from its PDF** **Given Density Function:** f(x)=frac1sqrt4picdot2e−frac(x−10)232quadtextfor−infty<x<inftyf(x) = \\frac{1}{\\sqrt{4\\pi}\\cdot 2} e^{-\\frac{(x-10)^2}{32}} \\quad \\text{for } -\\infty < x < \\infty **Step 1: Compare with the standard Normal PDF** The standard probability density function of a normal distribution is: f(x)=frac1sigmasqrt2pie−frac(x−mu)22sigma2f(x) = \\frac{1}{\\sigma \\sqrt{2\\pi}} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}} where: - mu\displaystyle \\mu is the Mean of the distribution. - sigma\displaystyle \\sigma is the Standard Deviation (textS.D.\displaystyle \\text{S.D.}) of the distribution. **Step 2: Identify the parameters** 1. Looking at the exponent term (x−mu)2=(x−10)2\displaystyle (x - \\mu)^2 = (x - 10)^2, we find the mean is: mu=10\\mu = 10 2. Looking at the exponent denominator 2sigma2=32impliessigma2=16impliessigma=4\displaystyle 2\\sigma^2 = 32 \\implies \\sigma^2 = 16 \\implies \\sigma = 4 (Standard Deviation). Thus: - The **Mean** is 10\displaystyle 10 (Option A). - The **Standard Deviation** is 4\displaystyle 4 (Option B). **Discrepancy Note:** The question asks for the **mean** of X\displaystyle X, which is mathematically 10\displaystyle 10 (Option A). However, the textbook answer key designates **Option B** (4\displaystyle 4) as the correct answer. Option B represents the standard deviation (sigma=4\displaystyle \\sigma = 4) of the distribution rather than its mean, indicating a typographical error in the question phrasing or the key. Since the question asks for the mean, Option A is the correct answer. Hence, **Option A** 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.

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A random variable X\displaystyle X follows Binomial Distribution With E(x)=2\displaystyle E(x)=2 and V(x)=1.2\displaystyle V(x)=1.2, then the value of n\displaystyle n is

When 'p\displaystyle p' is large than 0.5\displaystyle 0.5, the Binomial Distribution is:

For a Poisson distribution,

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