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 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.

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

For the Joint PMF as shown, find each of following quantities:𝑝𝑋 𝑥 , 𝑝𝑌 𝑦 , 𝑝𝑋|𝑌 𝑥 𝑦 , 𝑝𝑌|𝑋 𝑦 𝑥 , 𝐸[𝑋|𝑌 = 3]Also find whether 𝑋 and 𝑌 are independent or not.[The graph shows 𝑝𝑋,𝑌(𝑥, 𝑦)/12]

b) Find 𝑝𝑌 (𝑦), the marginal p.m.f. of 𝑌

Let the discrete RV 𝑋~𝑈[−2,2] (Uniform dist.). Let 𝑌 = 𝑋2a) What values X and Y can take? Find pdf’s of both X and Y.b) Compute the joint pdf, 𝑓𝑋𝑌(𝑥𝑖, 𝑦𝑖)c) Compute the E(X) and E(Y)d) Compute the Cov(X,Y)e) Compute the 𝜌𝑋𝑌 = 𝐶𝑜𝑟(𝑋, 𝑌).f) Are X and Y independent? Prove it.