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

Let X and Y be two random variables with joint probability mass function:P(X=i,Y=j) = (i+j)/36, for i,j=1,2,3,4What is the marginal probability mass function of X?Review LaterP(X=i) = 5(i+1)/36, for i=1,2,3,4P(X=i) = (5+i)/36, for i=1,2,3,4P(X=i) = 5/12, for i=1,2,3,4P(X=i) = (5i+1)/36, for i=1,2,3,4

Random variable X may take value -1 or 1; Y may take value 1, 2 or 3. The following Table shows the joint probability of random variables X and Y.Y1 2 3X -1 0.1 0.2 0.31 0.2 0.1 0.1Table 2: Joint probability distribution of X and Y. For example, the probability of X=-1 and Y=1 is 0.1. Provide solution to all questions below.(a) Marginal probability distribution of X(b) The expected value, variance and standard deviation of X. (c) Marginal probability distribution of Y(d) The expected value, variance and standard deviation of Y. (e) Conditional probability distribution of Y given X=-1.(f) The conditional expected value, conditional variance and conditional standard deviation of Y

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.

. 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 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)