Theoretical DistributionsMCQPYQ May 20Question 3413 of 230
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The variance of a binomial distribution with parameters n\displaystyle n and p\displaystyle p is:

Options

Anp2(1−p)\displaystyle np^2(1-p)
Bnp(1−p)\displaystyle \sqrt{np(1-p)}
Cnq(1−q)\displaystyle nq(1-q)
Dn2p(1−p)2\displaystyle n^2p(1-p)^2
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Correct Answer

✅ Option c — nq(1−q)\displaystyle nq(1-q)

All Options:

  • Anp2(1−p)\displaystyle np^2(1-p)
  • Bnp(1−p)\displaystyle \sqrt{np(1-p)}
  • Cnq(1−q)\displaystyle nq(1-q)
  • Dn2p(1−p)2\displaystyle n^2p(1-p)^2

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Detailed Solution & Explanation

**Variance of Binomial Distribution** For a Binomial distribution B(n,p)\displaystyle B(n, p), we have: - p\displaystyle p = probability of success - q=1−p\displaystyle q = 1 - p = probability of failure - n\displaystyle n = number of trials **Standard Formula:** Variance=npq=np(1−p)\text{Variance} = npq = np(1-p) **Checking each option:** **Option A:** np2(1−p)=np2q\displaystyle np^2(1-p) = np^2 q. This is NOT the variance formula (has an extra p\displaystyle p). **Option B:** np(1−p)=npq\displaystyle \sqrt{np(1-p)} = \sqrt{npq}. This is the **Standard Deviation**, not the variance. **Option C:** nq(1−q)\displaystyle nq(1-q) Since q=1−p\displaystyle q = 1 - p, we have 1−q=p\displaystyle 1 - q = p. Therefore: nq(1−q)=nq⋅p=npqnq(1-q) = nq \cdot p = npq This **equals** the variance formula npq\displaystyle npq. ✓ **Option D:** n2p(1−p)2=n2pq2\displaystyle n^2p(1-p)^2 = n^2pq^2. This is NOT the variance. **Conclusion:** Option C simplifies to npq\displaystyle npq, which is exactly the variance of a binomial distribution. 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

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

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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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