The sum and product of the mean and variance of a binomial distribution are 3.5 and 3 respectively. Find the binomial distribution.

Answer

To solve the problem, we need to find the parameters of a binomial distribution given the sum and product of its mean and variance. Let's denote the number of trials as \( n \) and the probability of success as \( p \). The probability of failure is \( q = 1 - p \).

### Step-by-Step Solution:

1. **Understand the Mean and Variance of Binomial Distribution**:
- The mean \( \mu \) of a binomial distribution is given by:
\[ \mu = np \]
- The variance \( \sigma^2 \) of a binomial distribution is given by:
\[ \sigma^2 = npq = np(1 - p) \]

2. **Set Up the Equations**:
- We are given that the sum of the mean and variance is 3.5:
\[ np + npq = 3.5 \]
- We are also given that the product of the mean and variance is 3:
\[ np \cdot npq = 3 \]

3. **Substituting Variance in Terms of Mean**:
- From the first equation, we can express \( npq \) in terms of \( np \):
\[ npq = 3.5 - np \]
- Substitute this into the second equation:
\[ np \cdot (3.5 - np) = 3 \]

4. **Rearranging the Equation**:
- This expands to:
\[ 3.5np - np^2 = 3 \]
- Rearranging gives us:
\[ np^2 - 3.5np + 3 = 0 \]

5. **Using the Quadratic Formula**:
- We can solve this quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 1 \), \( b = -3.5 \), and \( c = 3 \).
- The discriminant \( D \) is:
\[ D = (-3.5)^2 - 4 \cdot 1 \cdot 3 = 12.25 - 12 = 0.25 \]
- Therefore, the roots are:
\[ np = \frac{3.5 \pm \sqrt{0.25}}{2} = \frac{3.5 \pm 0.5}{2} \]
This gives us two possible values:
\[ np = \frac{4}{2} = 2 \quad \text{and} \quad np = \frac{3}{2} = 1.5 \]

6. **Finding Corresponding Variance Values**:
- For \( np = 2 \):
\[ npq = 3 \div 2 = 1.5 \]
- For \( np = 1.5 \):
\[ npq = 3 \div 1.5 = 2 \]

7. **Finding \( p \) and \( q \)**:
- We know \( q = 1 - p \). Let's consider \( np = 2 \):
- If \( np = 2 \) and \( npq = 1.5 \):
\[ q = \frac{1.5}{2} = 0.75 \implies p = 1 - 0.75 = 0.25 \]
- For \( np = 1.5 \):
- If \( np = 1.5 \) and \( npq = 2 \):
\[ q = \frac{2}{1.5} = \frac{4}{3} \text{ (not possible since } q \text{ cannot exceed } 1) \]

8. **Finding \( n \)**:
- Using \( np = 2 \) and \( p = 0.25 \):
\[ n = \frac{np}{p} = \frac{2}{0.25} = 8 \]

9. **Final Parameters**:
- Thus, we have:
\[ n = 8, \quad p = 0.25, \quad q = 0.75 \]

10. **Writing the Binomial Distribution**:
- The binomial distribution can be expressed as:
\[ P(X = r) = \binom{8}{r} (0.25)^r (0.75)^{8 - r} \]

### Final Answer:
The binomial distribution is given by:
\[ P(X = r) = \binom{8}{r} (0.25)^r (0.75)^{8 - r} \]