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edited Dec 18, 2018 at 8:12

If I am not mistaken, the probability distitributiondistribution pdf and the cummulative probability distribution cdf of $S^2$ should be given by

pdf = σ^2/n/σ^2 PDF[ChiSquareDistribution[n - 1]][t/n n/σ^2];
cdf = CDF[ChiSquareDistribution[n - 1]][t/n n/σ^2];

Some wuick consistency tests for the scalings:

Integrate[pdf, {t, 0, ∞}, Assumptions -> {σ > 0, n >= 2}] == 1
Limit[cdf, t -> ∞, Assumptions -> {σ > 0, n >= 2}] == 1
D[cdf, t] == pdf // Simplify

True

True

True

If I am not mistaken, the probability distitribution pdf and the cummulative probability distribution cdf of $S^2$ should be given by

pdf = σ^2/n PDF[ChiSquareDistribution[n - 1]][t/n σ^2];
cdf = CDF[ChiSquareDistribution[n - 1]][t/n σ^2];

Some wuick consistency tests for the scalings:

Integrate[pdf, {t, 0, ∞}, Assumptions -> {σ > 0, n >= 2}] == 1
Limit[cdf, t -> ∞, Assumptions -> {σ > 0, n >= 2}] == 1
D[cdf, t] == pdf // Simplify

True

True

True

If I am not mistaken, the probability distribution pdf and the cummulative probability distribution cdf of $S^2$ should be given by

pdf = n/σ^2 PDF[ChiSquareDistribution[n - 1]][t n/σ^2];
cdf = CDF[ChiSquareDistribution[n - 1]][t n/σ^2];

Some wuick consistency tests for the scalings:

Integrate[pdf, {t, 0, ∞}, Assumptions -> {σ > 0, n >= 2}] == 1
Limit[cdf, t -> ∞, Assumptions -> {σ > 0, n >= 2}] == 1
D[cdf, t] == pdf // Simplify

True

True

True

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created Dec 15, 2018 at 11:34

If I am not mistaken, the probability distitribution pdf and the cummulative probability distribution cdf of $S^2$ should be given by

pdf = σ^2/n PDF[ChiSquareDistribution[n - 1]][t/n σ^2];
cdf = CDF[ChiSquareDistribution[n - 1]][t/n σ^2];

Some wuick consistency tests for the scalings:

Integrate[pdf, {t, 0, ∞}, Assumptions -> {σ > 0, n >= 2}] == 1
Limit[cdf, t -> ∞, Assumptions -> {σ > 0, n >= 2}] == 1
D[cdf, t] == pdf // Simplify

True

True

True