Theoretical DistributionsMCQMTP May 19Question 3439 of 230
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The maximum value of the variance of binomial distribution with parameters n\displaystyle n and p\displaystyle p is

Options

An/2\displaystyle n/2
Bn/4\displaystyle n/4
Cnp(1−p)\displaystyle np(1-p)
D2n\displaystyle 2n
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Correct Answer

✅ Option b — n/4\displaystyle n/4

All Options:

  • An/2\displaystyle n/2
  • Bn/4\displaystyle n/4
  • Cnp(1−p)\displaystyle np(1-p)
  • D2n\displaystyle 2n

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Detailed Solution & Explanation

**Maximum Variance of Binomial Distribution**\n\nVariance of B(n,p)\displaystyle B(n, p) is:\nsigma2=np(1−p)=npq\\sigma^2 = np(1-p) = npq\n\n**Maximizing variance with respect to p:**\nfracddp[np(1−p)]=n(1−2p)=0\\frac{d}{dp}[np(1-p)] = n(1 - 2p) = 0\nRightarrowp=frac12\\Rightarrow p = \\frac{1}{2}\n\n**Second derivative test:**\nfracd2dp2[np(1−p)]=−2n<0quad(textconfirmsmaximum)\\frac{d^2}{dp^2}[np(1-p)] = -2n < 0 \\quad (\\text{confirms maximum})\n\n**Maximum variance (at } p = 1/2):**\nsigmamax2=ncdotfrac12cdotleft(1−frac12right)=ncdotfrac12cdotfrac12=fracn4\\sigma^2_{\\max} = n \\cdot \\frac{1}{2} \\cdot \\left(1 - \\frac{1}{2}\\right) = n \\cdot \\frac{1}{2} \\cdot \\frac{1}{2} = \\frac{n}{4}\n\nSo the **maximum value** of variance is dfracn4\displaystyle \\dfrac{n}{4}, achieved when p=dfrac12\displaystyle p = \\dfrac{1}{2}.\n\nHence, **Option B** 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

Variance

A measure of dispersion equal to the average of the squared deviations of each data point from the mean. Formula: σ² = Σ(xᵢ − x̄)² / n. Standard Deviation = √Variance.

More Questions from Theoretical Distributions

Find the variance of binomial distribution with n=10\displaystyle n=10, p=0.3\displaystyle p=0.3

When p=0.5\displaystyle p=0.5, the binomial distribution is

If mean and standard deviation of a binomial distribution is 10\displaystyle 10 and 2\displaystyle 2 respectively, q\displaystyle q will be

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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