Question

Given $X\sim B(n,p)$, if $E(X)=4$ and $\text{Var}(X)=2.4$, then $n=$

• A. $20$
• B. $15$
• C. $5$
• D. $10$

Hint:

In binomial problems, use both mean and variance equations together to eliminate variables.

Answer 1

The Correct Option is D

Step 1: Using formulas for binomial distribution.
For $X\sim B(n,p)$, \[ E(X)=np,\quad \text{Var}(X)=npq \]
Step 2: Substituting given values.
\[ np=4 \Rightarrow p=\frac{4}{n} \] \[ npq=2.4 \]
Step 3: Solving for $n$.
\[ 4\left(1-\frac{4}{n}\right)=2.4 \] \[ 4-\frac{16}{n}=2.4 \Rightarrow \frac{16}{n}=1.6 \Rightarrow n=10 \]
Step 4: Conclusion.
The value of $n$ is $10$.