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(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.

(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.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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(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(u|v). Are U, V independent? (f) Find a function g: R² → R² such that (X, Ỹ) = 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.
Transcribed Image Text:(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(u|v). Are U, V independent? (f) Find a function g: R² → R² such that (X, Ỹ) = 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.
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