Suppose that the joint PDF of (𝑋, 𝑌) is𝑓(𝑥, 𝑦) = 𝑐√1 − 𝑥2 − 𝑦2, 𝑥2 + 𝑦2 ≤ 1.Find the marginal PDF 𝑓𝑋(𝑥) and the constant 𝑐 . (iint: consider transformations like𝑦 = 𝑎 sin(𝜃) when calculating the integral) (10 points)

Let (X,Y) be a two-dimensional non-negative continuous random variable having the joint density𝑓(𝑥, 𝑦) = { 4𝑥𝑦𝑒−(𝑥2+𝑦2); 𝑥 ≥ 0, 𝑦 ≥ 00 𝑒𝑙𝑠𝑒 𝑤ℎ𝑒𝑟𝑒 Find the density function of U=√𝑋2 + 𝑌2.

eet the joint p.d.f. of X1 and X2 be:ℎ(𝑥1, 𝑥2) = {8𝑥1𝑥2 for 0 < 𝑥1 < 𝑥2 < 10 otherwisea) Find the joint p.d.f. of 𝑌1 = 𝑋1𝑋2and 𝑌2 = 𝑋2b) Are 𝑌1 and 𝑌2 independent? Why? (10 points

The joint pdf of two continuous random variables 𝑋X and 𝑌Y is given by𝑓𝑋𝑌(𝑥,𝑦)={4𝑥𝑦1440≤𝑥≤14,0≤𝑦≤140otherwisef XY​ (x,y)={ 14 4 4xy​ 0​ 0≤x≤14,0≤y≤14otherwise​ Are 𝑋X and 𝑌Y independent?YesNo

. The two-dimensional random variables ( 𝑋 , 𝑌 ) (X,Y) have the joint density function: 𝑓 ( 𝑥 , 𝑦 ) = 2 𝑥 + 𝑦 20 , 𝑥 = 0 , 1 , 2 ,  and  𝑦 = 0 , 1 , 2 f(x,y)= 20 2x+y ​ ,x=0,1,2, and y=0,1,2 (a) Find the joint cumulative distribution function (CDF) of 𝑋 X and 𝑌 Y. (b) Determine if 𝑋 X and 𝑌 Y are independent.

Suppose that X and Y have the joint probability density function: f(x,y) = 30x^2y^2(1 - x^2 - y^2) for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. What is the marginal probability density function of X?Review LaterfX(x) = 15x(1 - x^2) for 0 ≤ x ≤ 1fX(x) = 15x^2(1 - x^2) for 0 ≤ x ≤ 1fX(x) = 10x(1 - x^2) for 0 ≤ x ≤ 1fX(x) = 10x^2(1 - x^2) for 0 ≤ x ≤ 1