- Problem 1 A Find The Value Of A Such That The Following Function Is A Joint Density Of Two Random Variables F X Y 1 (76.34 KiB) Viewed 11 times
Problem 1*. (a) Find the value of A such that the following function is a joint density of two random variables: f(x, y) = Ae-x²-2y² (b) Let X, Y be the random variables admitting f from (a) as their joint density. Are X, Y independent? What is the distribution of each X, Y? (c) Let U, V be two random variables with joint density: fu,v (u, v) = Be-3u²+2uv-3v² Find B. (You may use the calculator for the rest of the problem.) (d) Find the marginal densities fu, fv. (e) Find the conditional fuv(ulv). Are U, V independent? (f) Find a function g: R² → R² such that (X, Y) = g(U, V) has joint density same as f from (a). Note: g is a linear transform. Hint: (u + v)² + 2(u-v)² = 3u² - 2uv + 3v². (g) Find the probability P(U > 1|V = 1). What is the distribution of (UV = 1)? (h) Find the probability P(3U +4V <0). Hint: aU + BV is a normal distribution.