For a binomial variate X with `n=6`, if `P(X=2)=9P(X=4)`, then its variance is

Options

A. `8//9`
B. `1//4`
C. `9//8`
D. 4

Answer

To solve the problem, we need to find the variance of a binomial random variable \( X \) with \( n = 6 \) given that \( P(X = 2) = 9 P(X = 4) \).

### Step-by-Step Solution:

1. **Understand the Binomial Probability Formula**:
The probability of getting exactly \( r \) successes in \( n \) trials for a binomial random variable is given by:
\[ P(X = r) = \binom{n}{r} p^r (1 - p)^{n - r} \]
where \( p \) is the probability of success and \( q = 1 - p \) is the probability of failure.

2. **Set Up the Equation**:
We are given:
\[ P(X = 2) = 9 P(X = 4) \]
Using the binomial formula:
\[ \binom{6}{2} p^2 (1 - p)^{4} = 9 \cdot \binom{6}{4} p^4 (1 - p)^{2} \]

3. **Simplify the Binomial Coefficients**:
Note that \( \binom{6}{2} = \binom{6}{4} \), so we can cancel these terms from both sides:
\[ p^2 (1 - p)^{4} = 9 p^4 (1 - p)^{2} \]

4. **Rearranging the Equation**:
Divide both sides by \( p^2 (1 - p)^{2} \) (assuming \( p \neq 0 \) and \( 1 - p \neq 0 \)):
\[ (1 - p)^{2} = 9 p^{2} \]

5. **Take the Square Root**:
Taking the square root of both sides gives:
\[ 1 - p = 3p \]

6. **Solve for \( p \)**:
Rearranging the equation:
\[ 1 = 4p \implies p = \frac{1}{4} \]

7. **Calculate \( q \)**:
Since \( q = 1 - p \):
\[ q = 1 - \frac{1}{4} = \frac{3}{4} \]

8. **Find the Variance**:
The variance \( \sigma^2 \) of a binomial distribution is given by:
\[ \sigma^2 = n p q \]
Substituting \( n = 6 \), \( p = \frac{1}{4} \), and \( q = \frac{3}{4} \):
\[ \sigma^2 = 6 \cdot \frac{1}{4} \cdot \frac{3}{4} = 6 \cdot \frac{3}{16} = \frac{18}{16} = \frac{9}{8} \]

### Final Answer:
The variance of the binomial random variable \( X \) is:
\[ \frac{9}{8} \]