4.5.6 Suppose that X1, …, Xn are iid N(μ, σ^2). Recall the Helmert variables Y1, …, Yn from the

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4.5.6 Suppose that $X_1, \dots, X_n$ are iid $N(\mu, \sigma^2)$. Recall the Helmert variables $Y_1, \dots, Y_n$ from the Example 4.4.9. (i) Find the pdf of $Y_i / Y_j$ for $i \neq j = 2, \dots, n$; (ii) Find the pdf of $Y_i / |Y_j|$ for $i \neq j = 2, \dots, n$; (iii) Find the pdf of $Y_i^2 / Y_j^2$ for $i \neq j = 2, \dots, n$; (iv) Find the pdf of $Y_i^2 / T$ where $T = \sum_{2 \le k \le n, k \neq i} Y_k^2$ for $i = 2, \dots, n$; (v) Find the pdf of $(Y_i^2 + Y_j^2) / U$ where $U = \sum_{2 \le k \le n, k \neq i, k \neq j} Y_k^2$ for $i \neq j = 2, \dots, n$. {Hint: Recall the distributional properties of the Helmert variables.}

          4.5.6 Suppose that $X_1, \dots, X_n$ are iid $N(\mu, \sigma^2)$. Recall the Helmert variables $Y_1, \dots, Y_n$ from the Example 4.4.9.
(i) Find the pdf of $Y_i / Y_j$ for $i \neq j = 2, \dots, n$;
(ii) Find the pdf of $Y_i / |Y_j|$ for $i \neq j = 2, \dots, n$;
(iii) Find the pdf of $Y_i^2 / Y_j^2$ for $i \neq j = 2, \dots, n$;
(iv) Find the pdf of $Y_i^2 / T$ where $T = \sum_{2 \le k \le n, k \neq i} Y_k^2$ for $i = 2, \dots, n$;
(v) Find the pdf of $(Y_i^2 + Y_j^2) / U$ where $U = \sum_{2 \le k \le n, k \neq i, k \neq j} Y_k^2$ for $i \neq j = 2, \dots, n$.
{Hint: Recall the distributional properties of the Helmert variables.}

4.5.6 Suppose that X1, …, Xn are iid N(μ, σ^2). Recall the Helmert variables Y1, …, Yn from the Example 4.4.9.
(i) Find the pdf of Yi / Yj for i ≠ j = 2, …, n;
(ii) Find the pdf of Yi / |Yj| for i ≠ j = 2, …, n;
(iii) Find the pdf of Yi^2 / Yj^2 for i ≠ j = 2, …, n;
(iv) Find the pdf of Yi^2 / T where T = ∑2 ≤ k ≤ n, k ≠ i Yk^2 for i = 2, …, n;
(v) Find the pdf of (Yi^2 + Yj^2) / U where U = ∑2 ≤ k ≤ n, k ≠ i, k ≠ j Yk^2 for i ≠ j = 2, …, n.
Hint: Recall the distributional properties of the Helmert variables.

Added by Patrick G.

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