Theoretical DistributionsMCQMTP Mar 21Question 3450 of 230
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If x\displaystyle x is binomial variate with parameter 15 and 1/3\displaystyle 1/3 what is the value of mode of the distribution.

Options

A5 and 6
B5.5
C5
D6
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Correct Answer

✅ Option c — 5

All Options:

  • A5 and 6
  • B5.5
  • C5
  • D6

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Detailed Solution & Explanation

**Mode of Binomial Distribution: n=15\displaystyle n=15, p=1/3\displaystyle p=1/3** **Formula for Mode:** For a Binomial distribution B(n,p)\displaystyle B(n, p), compute (n+1)p\displaystyle (n+1)p: (n+1)p=(15+1)timesfrac13=16timesfrac13=frac163=5.overline3(n+1)p = (15+1) \\times \\frac{1}{3} = 16 \\times \\frac{1}{3} = \\frac{16}{3} = 5.\\overline{3} Since (n+1)p=frac163\displaystyle (n+1)p = \\frac{16}{3} is **not an integer**, the distribution has a **unique mode**: textMode=lfloor(n+1)prfloor=lfloor5.33rfloor=5\\text{Mode} = \\lfloor (n+1)p \\rfloor = \\lfloor 5.33 \\rfloor = 5 **Note:** If (n+1)p\displaystyle (n+1)p were an integer, the distribution would have two modes: (n+1)p\displaystyle (n+1)p and (n+1)p−1\displaystyle (n+1)p - 1. Here, since 5.33\displaystyle 5.33 is not an integer, the unique mode is mathbf5\displaystyle \\mathbf{5}. 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

Mode

The value that appears most frequently in a data set. A data set can be unimodal (one mode), bimodal (two modes), or multimodal (more than two modes). It is the only measure of central tendency applicable to qualitative (categorical) data.

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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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