If `X~B(n,p),mu=20,sigma^(2)=10`, then find n and p.

Answer

To solve the problem, we need to find the values of \( n \) and \( p \) given that \( X \) follows a binomial distribution \( B(n, p) \) with mean \( \mu = 20 \) and variance \( \sigma^2 = 10 \).

### Step-by-step Solution:

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

2. **Setting Up the Equations**:
- From the mean, we have:
\[ n \cdot p = 20 \quad \text{(Equation 1)} \]
- From the variance, we have:
\[ n \cdot p \cdot (1 - p) = 10 \quad \text{(Equation 2)} \]

3. **Substituting Equation 1 into Equation 2**:
- From Equation 1, we know \( n \cdot p = 20 \). We can substitute this into Equation 2:
\[ 20 \cdot (1 - p) = 10 \]

4. **Solving for \( p \)**:
- Rearranging the equation gives:
\[ 20 - 20p = 10 \]
- Simplifying this results in:
\[ 20p = 20 - 10 \]
\[ 20p = 10 \]
\[ p = \frac{10}{20} = \frac{1}{2} \]

5. **Finding \( n \)**:
- Now that we have \( p = \frac{1}{2} \), we can substitute back into Equation 1 to find \( n \):
\[ n \cdot \frac{1}{2} = 20 \]
- Multiplying both sides by 2 gives:
\[ n = 40 \]

6. **Final Values**:
- Thus, we have:
\[ n = 40, \quad p = \frac{1}{2} \]

### Summary:
The values are:
- \( n = 40 \)
- \( p = \frac{1}{2} \)