Compare the corresponding algorithms

Question

Solve the following problem:

The noncentral chi-squared distribution,X²_p(λ) can be defined by

(i). a mixture representation (2.2), where g(x|y) is the density of and p(y) is the density X²_p+2 of P(λ/2), and

(ii). the sum of a X²_p-1  random variable and the square of a N (||θ||, 1).

a. Show that both those representations hold.

b. Show that the representations are equivalent if λ = θ2/2.

c. Compare the corresponding algorithms that can be derived from these representations among themselves and also with rchisq for small and large values of λ.