How is the chi-square distribution built from standard normal random variables?

## The definition Let `Z1, ..., Zk` be independent standard normal variables — mean 0, variance 1. Define `V = Z1^2 + Z2^2 + ... + Zk^2`. The distribution of `V` is called chi-square with `k` degrees of freedom. That is the definition; everything else about the distribution follows from it, which is why it is worth memorising the construction rather than the density formula. ## What the construction immediately tells you **Support.** A sum of squares cannot be negative, so `V` lives on `[0, infinity)`. Any answer that allows a negative chi-square value is self-refuting. **Mean.** For a standard normal, `E[Z^2] = Var(Z) + (E[Z])^2 = 1 + 0 = 1`. Summing `k` of them gives `E[V] = k`. The mean of a chi-square is its degrees of freedom. **Variance.** For a standard normal, `E[Z^4] = 3`, so `Var(Z^2) = 3 - 1 = 2`. Independence lets the variances add, giving `Var(V) = 2k`. Note the asymmetry with the mean: mean `k`, variance `2k`, so the standard deviation `sqrt(2k)` grows more slowly than the mean. **Shape.** With `k = 1` the distribution piles up near zero and has a long right tail — squaring sends most of a standard normal's mass into a small interval near 0. As `k` grows the shape becomes more symmetric; the skewness is `sqrt(8/k)`, which shrinks steadily. **Additivity.** If `V1` is chi-square on `d1` degrees of freedom and `V2` is an independent chi-square on `d2`, then `V1 + V2` is chi-square on `d1 + d2`. The proof is the construction itself: concatenating a list of `d1` squared normals with an independent list of `d2` squared normals produces a list of `d1 + d2` squared normals. Degrees of freedom add — they never multiply and never average. ## The t distribution as a ratio Suppose `Z` is standard normal and `V` is an independent chi-square on `k` degrees of freedom. Then `T = Z / sqrt(V / k)` has a t distribution with `k` degrees of freedom. Reading the formula tells you what the t distribution *is*: a standard normal whose scale has itself been made random by an independent, noisy estimate. The denominator `sqrt(V/k)` has mean near 1 but wobbles, and occasionally comes out small, which flings `T` far from zero. That is precisely why the t has heavier tails than the normal, and why the heaviness depends on `k`: as `k` grows, `V/k` concentrates near 1, the denominator stops wobbling, and the t approaches the standard normal. At the other extreme, `k = 1` gives the Cauchy distribution, which is also the ratio of two independent standard normals and has no mean at all. The t has finite variance `k/(k-2)` only when `k > 2`. ## The F distribution as a ratio of two chi-squares Let `V1` and `V2` be independent chi-squares on `d1` and `d2` degrees of freedom. Then `F = (V1 / d1) / (V2 / d2)` has an F distribution with numerator degrees of freedom `d1` and denominator degrees of freedom `d2`. Each chi-square is normalised by its own degrees of freedom first, so both numerator and denominator have mean 1 and the ratio hovers near 1 when the two sources of variation are comparable. Two structural facts follow from the construction: - the reciprocal of an F with `(d1, d2)` is an F with `(d2, d1)` — swapping the ratio swaps the pair; - the square of a t with `k` degrees of freedom is an F with `(1, k)`, since squaring `Z / sqrt(V/k)` produces `(Z^2 / 1) / (V / k)` and `Z^2` is a chi-square on one degree of freedom. The F has mean `d2 / (d2 - 2)` when `d2 > 2`, which is slightly above 1 and drifts toward 1 as the denominator degrees of freedom grow. ## Where the "degrees of freedom" number comes from The phrase counts independent squared normal contributions. When a quantity is computed from data after estimating something from the same data, one such contribution is used up: fixing an estimated centre removes one direction of free variation, so `n` observations can contribute only `n - 1` independent deviations around their own average. That accounting — how many free directions remain after the constraints imposed by estimation — is what determines the degrees of freedom parameter you carry into any of the three distributions above. ## Why it is worth knowing The chi-square, t and F are not three unrelated shapes to memorise; they are one normal-based construction viewed three ways: a sum of squares, a normal over a random scale, and a ratio of two random scales. Candidates who know the construction can reason about tail behaviour, about why degrees of freedom appear at all, and about limiting cases, instead of consulting a table. It is a differentiator rather than a screening question, but it separates candidates who understand the machinery from those who have only used it.