Question

The mean of a binomial distribution is \(5\) and its standard deviation is \(2\). Then the value of \(n\) and \(p\) are

• A. \(\left(\frac{4}{5},25\right)\)
• B. \(\left(25,\frac{4}{5}\right)\)
• C. \(\left(\frac{1}{5},25\right)\)
• D. \(\left(25,\frac{1}{5}\right)\)

Hint:

For binomial distribution, use: \[ \mu=np,\qquad \sigma^2=npq \] First find \(q\), then \(p\), and finally \(n\).

Answer 1

The Correct Option is D

Concept:
For a binomial distribution with parameters \(n\) and \(p\): Mean is: \[ \mu=np \] Variance is: \[ \sigma^2=npq \] where: \[ q=1-p \] Standard deviation is: \[ \sigma=\sqrt{npq} \]

Step 1: Write the given values.
Mean: \[ \mu=5 \] Standard deviation: \[ \sigma=2 \] So variance is: \[ \sigma^2=2^2=4 \]

Step 2: Use the mean formula.
For binomial distribution: \[ \mu=np \] So: \[ np=5 \]

Step 3: Use the variance formula.
Variance: \[ \sigma^2=npq \] So: \[ npq=4 \] But from mean: \[ np=5 \] Therefore: \[ 5q=4 \] \[ q=\frac{4}{5} \]

Step 4: Find \(p\).
Since: \[ p+q=1 \] \[ p=1-q \] \[ p=1-\frac{4}{5} \] \[ p=\frac{1}{5} \]

Step 5: Find \(n\).
Using: \[ np=5 \] Substitute: \[ p=\frac{1}{5} \] \[ n\cdot\frac{1}{5}=5 \] \[ n=25 \] Hence, the correct answer is: \[ \boxed{(D)\ \left(25,\frac{1}{5}\right)} \]