NIMCET 2012 — Mathematics PYQ

For any random variable $X$ that is bounded by the interval $[a,b]$, there is a mathematical upper bound for its variance. This is known as Popoviciu's inequality.

2. The Formula:

The variance of $X$ satisfies the following condition:

3. Analysis of Options:

• Since $\text{Var}(X)\le \frac{(b-a{)}^2}{4}$, and we know that $\frac{(b-a{)}^2}{4}$ is strictly less than or equal to $(b-a{)}^2$:

  $$\text{Var}(X)\le \frac{(b-a{)}^2}{4}\le (b-a{)}^2$$

• Therefore, the inequality $(b-a{)}^2\ge \text{Var}(X)$ is always true for a bounded random variable.

Conclusion:

Option (a) correctly identifies a valid upper bound for the variance given the range $[a,b]$.

Correct Option: (a)