In a binomial distribution, the mean is 4 and variance is 3. Then its mode is

Options

A. 6
B. 5
C. 4
D. 7

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We are given that in a binomial distribution:
- Mean (μ) = 4
- Variance (σ²) = 3

### Step 2: Use the formulas for mean and variance in a binomial distribution
In a binomial distribution, the mean (μ) is given by:
\[ \mu = n \cdot p \]
And the variance (σ²) is given by:
\[ \sigma^2 = n \cdot p \cdot q \]
where \( q = 1 - p \).

### Step 3: Set up the equations
From the mean, we have:
\[ n \cdot p = 4 \quad \text{(Equation 1)} \]
From the variance, we have:
\[ n \cdot p \cdot q = 3 \quad \text{(Equation 2)} \]

### Step 4: Substitute \( q \) in terms of \( p \)
Since \( q = 1 - p \), we can substitute this into Equation 2:
\[ n \cdot p \cdot (1 - p) = 3 \]

### Step 5: Divide Equation 2 by Equation 1
Dividing Equation 2 by Equation 1 gives:
\[ \frac{n \cdot p \cdot (1 - p)}{n \cdot p} = \frac{3}{4} \]
This simplifies to:
\[ 1 - p = \frac{3}{4} \]
Thus, we find:
\[ p = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 6: Substitute \( p \) back to find \( n \)
Now, substitute \( p \) back into Equation 1:
\[ n \cdot \frac{1}{4} = 4 \]
Multiplying both sides by 4 gives:
\[ n = 16 \]

### Step 7: Find the mode of the binomial distribution
The mode \( R \) of a binomial distribution can be found using the formula:
\[ n \cdot p \]
The mode is given by:
\[ \lfloor (n + 1) \cdot p \rfloor \text{ or } \lfloor (n + 1) \cdot p \rfloor - 1 \]
Calculating:
\[ (n + 1) \cdot p = (16 + 1) \cdot \frac{1}{4} = 17 \cdot \frac{1}{4} = \frac{17}{4} = 4.25 \]

### Step 8: Determine the integral values for the mode
Thus, we have:
\[ 3.25 < R < 4.25 \]
The only integer value in this range is:
\[ R = 4 \]

### Conclusion
Therefore, the mode of the binomial distribution is:
\[ \text{Mode} = 4 \]