If the mean and variance of a binomial variate X are `(4)/(3),(8)/(9)` respectively , then P(X = 2) =

Options

A. `(4)/(27)`
B. `(8)/(81)`
C. `(16)/(81)`
D. `(8)/(27)`

Answer

To solve the problem, we need to find \( P(X = 2) \) for a binomial random variable \( X \) with given mean and variance.

### Step-by-Step Solution:

1. **Identify the Mean and Variance:**
- Given:
\[ \text{Mean} = \frac{4}{3}, \quad \text{Variance} = \frac{8}{9} \]

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

3. **Express \( q \) in terms of \( p \):**
- From the variance formula, we can express \( q \) as:
\[ q = 1 - p \]

4. **Substitute the Mean into the Variance Formula:**
- From the mean, we have:
\[ n \cdot p = \frac{4}{3} \]
- Substitute \( n \cdot p \) into the variance equation:
\[ n \cdot p \cdot q = \frac{8}{9} \]
- Substitute \( n \cdot p = \frac{4}{3} \):
\[ \frac{4}{3} \cdot q = \frac{8}{9} \]

5. **Solve for \( q \):**
- Rearranging gives:
\[ q = \frac{8}{9} \cdot \frac{3}{4} = \frac{2}{3} \]

6. **Find \( p \):**
- Since \( p + q = 1 \):
\[ p = 1 - q = 1 - \frac{2}{3} = \frac{1}{3} \]

7. **Substitute \( p \) back to find \( n \):**
- Substitute \( p \) into the mean equation:
\[ n \cdot \frac{1}{3} = \frac{4}{3} \]
- Solving for \( n \):
\[ n = 4 \]

8. **Calculate \( P(X = 2) \):**
- Use the binomial probability formula:
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]
- For \( r = 2 \):
\[ P(X = 2) = \binom{4}{2} \left(\frac{1}{3}\right)^2 \left(\frac{2}{3}\right)^{4-2} \]

9. **Calculate \( \binom{4}{2} \):**
- Compute:
\[ \binom{4}{2} = \frac{4!}{2! \cdot 2!} = \frac{4 \cdot 3}{2 \cdot 1} = 6 \]

10. **Combine the calculations:**
\[ P(X = 2) = 6 \cdot \left(\frac{1}{3}\right)^2 \cdot \left(\frac{2}{3}\right)^2 \]
\[ = 6 \cdot \frac{1}{9} \cdot \frac{4}{9} = 6 \cdot \frac{4}{81} = \frac{24}{81} = \frac{8}{27} \]

### Final Answer:
\[ P(X = 2) = \frac{8}{27} \]