Let (𝑋, 𝑌 ) be a discrete bivariate random vector with joint p.m.f. given by𝑓𝑋,𝑌 (𝑥, 𝑦) =(𝑐 · 𝑥𝑦 if 𝑥 ∈ {1, 2, 3}, 𝑦 ∈ {1, 2, 3}0 otherwisewhere 𝑐 > 0 is an as-of-yet undetermined constant.(a) Find the value of 𝑐

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

Suppose that (𝑋, 𝑌) follows a bivariate normal distribution 𝑁(𝜇1, 𝜇2, 𝜎12, 𝜎22, 𝜌) with𝜇1 = 𝜇2 = 0 , 𝜎12 = 1 , 𝜎22 = 2 , and 𝜌 = 1/√2. eet 𝑈 = 𝑎𝑋 + 𝑏𝑌 , 𝑉 = 𝑐𝑋 + 𝑑𝑌 , findthe four equations that constants 𝑎, 𝑏, 𝑐, 𝑑 need to satisfy such that 𝑈 and 𝑉 areindependent standard normal random variables (no need to solve the equations). (10 points

(30 points) Suppose Company 1's stock return 𝑋𝑋 is a random variable and takes three possiblevalues: {-0.1, 0.1, 0.2}. And Company 2's stock return 𝑌𝑌 is a random variable and takes twopossible values: {-0.3, 0.4}. The joint probability distribution 𝑓𝑓(𝑋𝑋, 𝑌𝑌) is given as follows:𝑓𝑓(−0.1, −0.3) = 0.1, 𝑓𝑓(0.1, −0.3) = 0.2, 𝑓𝑓(0.2, −0.3) = 0.2,𝑓𝑓(−0.1,0.4) = 0.2, 𝑓𝑓(0.1,0.4) = 0.2, 𝑓𝑓(0.2,0.4) = 0.1.Please calculate the following:(a) Marginal distributions: 𝑓𝑓𝑋𝑋(𝑥𝑥) and 𝑓𝑓𝑌𝑌(𝑦𝑦). (4 points)(b) Mean: 𝐸𝐸(𝑋𝑋) and 𝐸𝐸(𝑌𝑌). (4 points)(c) Variance: 𝑉𝑉𝑎𝑎𝑟𝑟(𝑋𝑋) and 𝑉𝑉𝑎𝑎𝑟𝑟(𝑌𝑌). (4 points)(d) Covariance: 𝐶𝐶𝐶𝐶𝐶𝐶(𝑋𝑋, 𝑌𝑌). (2 points)(e) Conditional expectations: 𝐸𝐸(𝑋𝑋|𝑌𝑌 = −0.3) and 𝐸𝐸(𝑋𝑋|𝑌𝑌 = 0.4). (8 points)(f) Conditional variances: 𝑉𝑉𝑎𝑎𝑟𝑟(𝑋𝑋|𝑌𝑌 = −0.3) and 𝑉𝑉𝑎𝑎𝑟𝑟(𝑋𝑋|𝑌𝑌 = 0.4). (8 points)

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

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]