Question

In a Binomial distribution \(B(n,p)\), if the mean and variance are \(15\) and \(10\) respectively, then the value of the parameter \(n\) is

• A. \(28\)
• B. \(16\)
• C. \(45\)
• D. \(25\)

Hint:

For a Binomial distribution, remember the standard formulas: \(\text{Mean}=np\) and \(\text{Variance}=npq\), where \(q=1-p\).

Answer 1

The Correct Option is C

Step 1: Write the basic formulas.
For $B(n,p)$, mean $=np$ and variance $=npq$ with $q=1-p$.
Step 2: Use the given mean.
The mean is $15$, so $np=15$.
Step 3: Use the given variance.
The variance is $10$, so $npq=10$.
Step 4: Find $q$.
Dividing, $\dfrac{npq}{np}=\dfrac{10}{15}$, so $q=\dfrac{2}{3}$.
Step 5: Find $p$.
Since $p=1-q=1-\dfrac{2}{3}=\dfrac{1}{3}$.
Step 6: Find $n$.
From $np=15$ with $p=\dfrac{1}{3}$, we get $n=15\times 3=45$.
\[ \boxed{45} \]