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In a Binomial Distribution, if p, q and n are probability of success, failure and number of trials respectively then variance is given by
- a)np
- b)npq
- c)np2q
- d)npq2
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
In a Binomial Distribution, if p, q and n are probability of success, ...
For a discrete probability function, the variance is given by
Variance(V) = 
Where µ is the mean, substitute P(x)=nCx px q(n-x) in the above equation and put µ = np to obtain
V = npq.
Most Upvoted Answer
In a Binomial Distribution, if p, q and n are probability of success, ...
Understanding Variance in Binomial Distribution
The Binomial Distribution describes the number of successes in a fixed number of independent Bernoulli trials, each yielding a success with probability p and a failure with probability q (where q = 1 - p). The formula for variance is crucial for understanding the distribution's spread.
Key Components of Binomial Distribution
- n: Number of trials
- p: Probability of success on a single trial
- q: Probability of failure, defined as q = 1 - p
Variance Formula
The variance (Var) of a Binomial Distribution is calculated using the formula:
- Variance = npq
This formula indicates how much the number of successes can vary from the expected value (mean).
Explanation of the Formula
- np: Represents the expected number of successes in n trials.
- q: Since q is the probability of failure, multiplying by q accounts for the variability in the number of successes due to failures.
This means the variance is influenced by both the number of trials and the probabilities of success and failure.
Why Option B is Correct
Option B (npq) correctly encapsulates the relationship between the number of trials and the probabilities of success and failure. Other options (np, np^2, npq^2) do not accurately represent the variability in the context of a Binomial Distribution.
Conclusion
Understanding the variance in a Binomial Distribution is essential for statistical analysis. The formula npq effectively captures the spread of data points around the mean, making option B the correct answer.
The Binomial Distribution describes the number of successes in a fixed number of independent Bernoulli trials, each yielding a success with probability p and a failure with probability q (where q = 1 - p). The formula for variance is crucial for understanding the distribution's spread.
Key Components of Binomial Distribution
- n: Number of trials
- p: Probability of success on a single trial
- q: Probability of failure, defined as q = 1 - p
Variance Formula
The variance (Var) of a Binomial Distribution is calculated using the formula:
- Variance = npq
This formula indicates how much the number of successes can vary from the expected value (mean).
Explanation of the Formula
- np: Represents the expected number of successes in n trials.
- q: Since q is the probability of failure, multiplying by q accounts for the variability in the number of successes due to failures.
This means the variance is influenced by both the number of trials and the probabilities of success and failure.
Why Option B is Correct
Option B (npq) correctly encapsulates the relationship between the number of trials and the probabilities of success and failure. Other options (np, np^2, npq^2) do not accurately represent the variability in the context of a Binomial Distribution.
Conclusion
Understanding the variance in a Binomial Distribution is essential for statistical analysis. The formula npq effectively captures the spread of data points around the mean, making option B the correct answer.
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