principle of mathematical statistics

Problem 3 Let X1, ..., X,, be independent and identically distributed continuous random variables with a positive continuous joint probability density function f(x1. . .., X,). (i) Suppose that the distribution of X1. .... X, is radially symmetric about the origin, which means that the joint probability density function f satisfies f(x1 , . .., Xn) = f()'1. . . ., Vn) whenever yi .. . + )n. What are all possible distributions of X1? (You can specify the form of the density function if you like, but make sure that you only specify valid density functions.) Explain your answer. (ii) Suppose that the joint probability density function satisfies the relation f ( x 1 , . . . . X,) = f()'1, . . ..)'n) whenever Ixil + ... + xml = built ... + lynl. What are all possible distributions of X1? (You can specify the form of the density function if you like, but make sure that you only specify valid density functions.) Explain your