Let X1 X2 be independent r vs distributed as N0 sigma2Then s

Let X_1, X_2, be independent r. v.\'s distributed as N(0, sigma^2).Then show that: The r. v. x^2 1 + x^2 2 has the Negative Exponential distribution with parameter lambda = 1/2 sigma^2; The r. v. X_1/x_2 has the Cauchy distribution with mu = 0 and sigma = 1; The r. v.\' s x^2 I +x^2 2 and X_1/x_2 are independent need go line (iii) just

Solution

iii)

X1^2 + X2^2

mean = 0 +0 = 0

var = sigma^2   + sigma^2 = 2 * sigma^2

X1 / X2

mean = 0

var =    Sigma / Sigma = 1

then there are independent