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

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

3.  Suppose the random variable X𝑋 is normally distributed with μ=50𝜇=50 and σ2=36𝜎2=36 . What is the value of b𝑏 such that P(X≥b)=0.3264P(𝑋≥𝑏)=0.3264 ?Multiple choice 1 Question 3  45.8  59.1  52.7  47.3

eet 𝑋 be a continuous random variable with PDF that is symmetric about zero, and let𝑌 = 𝑆𝑋 , where 𝑆 is a discrete random variable independent of 𝑋 and has PMF𝑃(𝑆 = 1) = 𝑃(𝑆 = −1) = 0.5. Compute Cov(𝑋, 𝑌). Are 𝑋 and 𝑌 independent? Justifyyour answer. (10 points

4.  Suppose the random variable X𝑋 is normally distributed with μ=50𝜇=50 and σ2=16𝜎2=16 . What is P(49≤X≤55)P(49≤𝑋≤55) ?Multiple choice 2 Question 3  0.3756  0.4931  0.6734  0.2098

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)