ChiSquare, Cauchy and T-Distributions from Normal | UPSC ISS 2024 Paper-1 | Problem-8, 9, 10

This short deals with expectations and distributions involving sums and ratios of independent standard normal variables \(X_i \sim N(0,1)\). The sum of squares of four such variables follows a chi-square distribution with 4 degrees of freedom with expectation 4. The ratio involving linear sums has expectation zero due to independence and zero mean. The ratio of sums scaled appropriately forms a Cauchy distribution with a calculated scale parameter, which is adjusted to standard Cauchy by finding the scaling factor \( \lambda = \sqrt{\frac{2}{5}} \). This explanation clarifies advanced distribution relationships important for statistics learners and UPSC aspirants.

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