Question

The discrete random variables \(X\) and \(Y\) are independent from one another and are defined as \(X \sim B(n_1, 0.5)\) and \(Y \sim B(n_2, 0.4)\). If the variance of both \(X\) and \(Y\) is 6, then \(\sqrt{n_1+n_2} =\)

• A. 7
• B. 6
• C. 5
• D. 4

Hint:

For independent binomial random variables, use \(\text{Var}(X) = n p (1-p)\) to find unknown \(n\) or \(p\), then combine as needed.

Answer 1

The Correct Option is A

Step 1: Variance formula for binomial.
For \(X \sim B(n_1, p_1)\), \(\text{Var}(X) = n_1 p_1 (1-p_1)\).
For \(Y \sim B(n_2, p_2)\), \(\text{Var}(Y) = n_2 p_2 (1-p_2)\).

Step 2: Apply the given variances.
\[ n_1 \cdot 0.5 \cdot 0.5 = 6 \implies n_1 = \frac{6}{0.25} = 24 \]
\[ n_2 \cdot 0.4 \cdot 0.6 = 6 \implies n_2 = \frac{6}{0.24} = 25 \]

Step 3: Sum of \(n_1 + n_2\).
\[ n_1 + n_2 = 24 + 25 = 49 \]

Step 4: Take square root.
\[ \sqrt{n_1 + n_2} = \sqrt{49} = 7 \]

Step 5: Final conclusion.
Hence, \[ \boxed{7} \]