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]

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

4Q. The joint probability mass function of discrete random variables 𝑋 X and 𝑌 Y is given by: 𝑃 [ 𝑋 = 𝑥 , 𝑌 = 𝑦 ] = { 𝑘 ( 4 𝑥 + 𝑦 ) , 𝑥 = 1 , 2 , 3  and  𝑦 = 1 , 2 , 3 0 , otherwise P[X=x,Y=y]={ k(4x+y), 0, ​ x=1,2,3 and y=1,2,3 otherwise ​ Find the value of 𝑘 k. Then, calculate 𝑃 ( 𝑋 = 2  and  𝑌 = 3 ) P(X=2 and Y=3).

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

The joint distribution of 𝑋X and 𝑌Y is given by                             𝑓𝑋𝑌(𝑥,𝑦)=916×4𝑥+𝑦,f XY​ (x,y)= 16×4 x+y 9​ ,where 𝑇𝑋,𝑇𝑌∈{0,1,2,…}.T X​ ,T Y​ ∈{0,1,2,…}.1 pointFind the probability mass function of 𝑋+𝑌X+Y.𝑘916⋅4𝑘k 16⋅4 k 9​ (𝑘+1)916⋅4𝑘(k+1) 16⋅4 k 9​ (𝑘+1)916⋅4𝑘+1(k+1) 16⋅4 k+1 9​ 𝑘916⋅4𝑘+1k 16⋅4 k+1 9​

. 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.