AP EAMCET · Maths · Statistics
The random variable takes the values \(1,2,3\), \(\ldots, m\). If \(P(X=n)=\frac{1}{m}\) to each \(n\), then the variance of \(X\) is
- A \(\frac{(m+1)(2 m+1)}{6}\)
- B \(\frac{m^2-1}{12}\)
- C \(\frac{m+1}{2}\)
- D \(\frac{m^2+1}{12}\)
Answer & Solution
Correct Answer
(B) \(\frac{m^2-1}{12}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \operatorname{var}(X)=\sum_{i=2}^{\infty} P_i\left(X_i-\bar{X}\right)^2 \\ & \bar{X}=\frac{1+2+\ldots+m}{m}=\frac{m(m+1)}{2 \cdot m}=\frac{m+1}{2} \\ & \operatorname{var}(X)=\{P(X=1)+P(X=2)+\ldots+P(X=m)\} \\ &…
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