Question

The mean of a binomial distribution is 5, then its variance is

• A. \(>\) 5
• B. 5
• C. \(<\) 5
• D. 25

Hint:

Remember that for a binomial distribution, the variance is always less than or equal to the mean (\(\sigma^2 = \mu(1 - p)\), and since \(0 \le 1 - p \le 1\), then \(\sigma^2 \le \mu\)). The variance is equal to the mean only when \(p = 0\) or \(p = 1\), which corresponds to degenerate binomial distributions. For a proper binomial distribution where \(0 < p < 1\), the variance is strictly less than the mean.

Answer 1

The Correct Option is C

Step 1: Recall the formulas for the mean and variance of a binomial distribution.
A binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. Let \(n\) be the number of trials and \(p\) be the probability of success in each trial.
Mean (\(\mu\)): The mean of a binomial distribution is given by the formula:
$$\mu = np$$- Variance (\(\sigma^2\)): The variance of a binomial 
distribution is given by the formula:$$\sigma^2 = np(1 - p)$$ 
Step 2: Use the given information about the mean.
We are given that the mean of the binomial distribution is 5: $$\mu = np = 5$$ 
Step 3: Express the variance in terms of the mean and the probability of success.
We can rewrite the variance formula by substituting \(np = 5\): $$\sigma^2 = (np)(1 - p) = 5(1 - p)$$ 
Step 4: Analyze the possible values of the probability of success \(p\).
The probability of success \(p\) in a Bernoulli trial must be between 0 and 1 (inclusive):
$$0 \le p \le 1$$
Step 5: Determine the range of possible values for \(1 - p\).
If \(0 \le p \le 1\), then by multiplying by -1 and adding 1, we get: $$0 \le 1 - p \le 1$$ 
Step 6: Determine the range of possible values for the variance \(\sigma^2\).
Since \(\sigma^2 = 5(1 - p)\) and \(0 \le 1 - p \le 1\), we can multiply the inequality by 5: $$5 \times 0 \le 5(1 - p) \le 5 \times 1$$ $$0 \le \sigma^2 \le 5$$ However, for a valid binomial distribution, we must have \(0<p<1\), because if \(p = 0\) or \(p = 1\), the variance would be 0, which typically isn't the scenario implied in such questions unless specified. If \(0<p<1\), then \(0<1 - p<1\), and consequently: $$0<\sigma^2<5$$ This means that the variance of the binomial distribution must be less than 5. 
Step 7: Compare the result with the given options.
The result that the variance \(\sigma^2\) is less than 5 matches option (3).