Random Variable And Variance

The variance of a discrete random variable is Var(X) = E(X²) − [E(X)]², the expected square minus the square of the expectation, capturing dispersion about the mean. Computing both moments for a fair die is a standard JEE Advanced exercise. Each face 1 through 6 occurs with probability 1/6, so E(X) = (1 + 2 + 3 + 4 + 5 + 6)/6 = 21/6 = 7/2. Next E(X²) = (1 + 4 + 9 + 16 + 25 + 36)/6 = 91/6. Therefore Var(X) = 91/6 − (7/2)² = 91/6 − 49/4. Taking a common denominator 12 gives 182/12 − 147/12 = 35/12. Option 7/2 mistakes the mean for the variance. Option 91/6 reports E(X²) without subtracting the squared mean. Option 17/12 results from arithmetic slips in the subtraction. The result rests on the variance identity Var(X) = E(X²) − [E(X)]², which is algebraically equivalent to averaging the squared deviations from the mean but is usually faster to apply because the two raw moments are easy to tabulate. Computing the deviation form directly would require subtracting 7/2 from each face, squaring, and averaging, and it yields the identical 35/12, providing an independent confirmation. Plausibility check: 35/12 ≈ 2.92 is positive as every variance must be, and the standard deviation √(35/12) ≈ 1.71 is a reasonable spread for outcomes ranging over the six values 1 to 6, confirming overall consistency.