3) Rotational invariance and correlation: Y ~ unif(Bn) and U ~ unif(Sn-1) a) With Xi = U2i, determine EX1, s(X1) and r(X1, X2)

3) Rotational invariance and correlation: Y ~ unif(Bⁿ) and U ~ unif(S^n-1)

a) With X_i = U²_i, determine EX₁, s(X₁) and r(X₁, X₂).

b) With V_i = Y²_i, determine EV₁, s(V₁) and r(V₁, V₂).

5) Order statistics with the standard exponential: Z₁,……., Z_n IID Z ~ exp(1), n Î N

a) With Z₀ = 0, determine the joint distribution of the coverage, Z_(i) – Z_(i-1), I = 1, …., n.

b) using the result in a), or otherwise, obtain EZ_(i) and s(Z_(i)) for every 1 £ i £ n as well as the correlation coefficients r(Z_(i), Z_(j)) for 1 £ i < j £ n.