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: Get n and p
$q=0.75$, $p=0.25$, $n=4$.

Step 2: Ratio test
$\dfrac{P(1)}{P(2)}=\dfrac{4pq^3}{6p^2q^2}=\dfrac{2q}{3p}=\dfrac{2(0.75)}{3(0.25)}=2$. So $P(1)=2P(2)$, option (B).

Final Answer:
Option (B) holds for $n=4$, $p=\frac14$. \[ \boxed{P(X=1)=2P(X=2)} \]