Question

A random variable \(X\) follows binomial distribution with mean \(\alpha\) and variance \(\beta\). Then

• A. \(0<\alpha<\beta\)
• B. \(0<\beta<\alpha\)
• C. \(\alpha<0<\beta\)
• D. \(\beta<0<\alpha\)

Hint:

For binomial distribution, variance \(=npq\) is always less than mean \(=np\) because \(0<q<1\).

Answer 1

The Correct Option is B

Step 1: Recall mean and variance of binomial distribution.
If \(X\sim Bin(n,p)\), then: 
\[ \alpha = np \] \[ \beta = npq = np(1-p) \] 
Step 2: Compare \(\alpha\) and \(\beta\). 
\[ \beta = np(1-p) = \alpha(1-p) \] 
Since \(0 \[ 0<1-p<1 \Rightarrow \beta = \alpha(1-p)<\alpha \] 
Also both are positive: 
\[ \alpha>0,\ \beta>0 \] 
Step 3: Conclusion. 
\[ 0<\beta<\alpha \] 
Final Answer: 
\[ \boxed{0<\beta<\alpha} \]