If a binomial distribution has mean 2.4 and variance is 1.44, then n=

Options

A. 10
B. 6
C. 16
D. 20

Answer

To solve the problem, we need to find the value of \( n \) in a binomial distribution given the mean and variance. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understand the Formulas**:
The mean \( \mu \) and variance \( \sigma^2 \) of a binomial distribution can be expressed as:
\[ \text{Mean} (\mu) = n \cdot p \]
\[ \text{Variance} (\sigma^2) = n \cdot p \cdot q \]
where \( q = 1 - p \).

2. **Given Values**:
From the problem, we know:
\[ \mu = 2.4 \quad \text{and} \quad \sigma^2 = 1.44 \]

3. **Set Up the Equations**:
From the mean:
\[ n \cdot p = 2.4 \quad \text{(1)} \]
From the variance:
\[ n \cdot p \cdot q = 1.44 \quad \text{(2)} \]

4. **Express \( q \) in Terms of \( p \)**:
Since \( q = 1 - p \), we can substitute \( q \) in equation (2):
\[ n \cdot p \cdot (1 - p) = 1.44 \]

5. **Substitute \( n \cdot p \) from Equation (1)**:
From equation (1), we have \( n \cdot p = 2.4 \). Substitute this into the modified equation (2):
\[ 2.4 \cdot (1 - p) = 1.44 \]

6. **Solve for \( p \)**:
Rearranging gives:
\[ 1 - p = \frac{1.44}{2.4} \]
Calculate \( \frac{1.44}{2.4} \):
\[ 1 - p = 0.6 \]
Therefore:
\[ p = 1 - 0.6 = 0.4 \]

7. **Substitute \( p \) Back to Find \( n \)**:
Now substitute \( p \) back into equation (1):
\[ n \cdot 0.4 = 2.4 \]
Solving for \( n \):
\[ n = \frac{2.4}{0.4} = 6 \]

8. **Conclusion**:
The value of \( n \) is \( 6 \).

### Final Answer:
The value of \( n \) is \( 6 \).