Question

If the mean and the variance of a binomial variate X are 1 and 0.75 respectively, then which of the following is true?

• A. \(P(X = 0) = 3P(X = 4)\)
• B. \(P(X = 1) = 2P(X = 2)\)
• C. \(P(X = 3) = 3P(X = 4)\)
• D. \(3P(X = 0) = 4P(X = 1)\)

Hint:

Find \(n\) and \(p\) from \(np=1\) and \(npq=0.75\), then compare probabilities.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
For a binomial variate, mean \(=np\) and variance \(=npq\).

Step 2: Key Formula or Approach
\(q=\dfrac{npq}{np}=0.75\), so \(p=0.25\) and \(n=\dfrac{1}{0.25}=4\).

Step 3: Detailed Explanation
\(P(X=r)=\binom4r\left(\tfrac14\right)^r\left(\tfrac34\right)^{4-r}=\dfrac{\binom4r3^{4-r}}{256}\).
\(P(0)=\dfrac{81}{256}\), \(P(1)=\dfrac{4\cdot27}{256}=\dfrac{108}{256}\), \(P(2)=\dfrac{6\cdot9}{256}=\dfrac{54}{256}\), \(P(3)=\dfrac{12}{256}\), \(P(4)=\dfrac1{256}\).
Now \(P(1)=108\) and \(2P(2)=108\), so \(P(X=1)=2P(X=2)\), which is option (B).

Final Answer:
The true relation is \(P(X=1)=2P(X=2)\), option (B). \[ \boxed{P(X=1)=2P(X=2)\ \text{(B)}} \]