2. From a group of three Republicans, two Democrats, and one independent, a committee of two people is to be randomly selected. Let Y₁ denote the number of Republicans and Y2 denote the number of Democrats on the committee. a) Find the sample space associated with the experiment. b) Find the joint probability function of Y₁ and Y2. c) Find the marginal probability function of Y₂. d) Find the conditional distribution of Y₁ given that Y₂ = 1. e) Is the number of Republicans in the sample independent of the number of Democrats? (Is Y₁ independent of Y₂?) f) Find E(Y₁- Y₂). g) Find V(Y₁₂). 3. Suppose that Y, and Y2 have join probability density function f(y1 y2) = ke-6y1-3y2, for y₁ > 0 and y20, and f(y1, y2) = 0 elsewhere. e) Find k. f) Find P(y1 y2/2). g) Find P(max(y1, y2) ≤ 1). h) Find the conditional density of Y₁, given Y₂ = y2. i) Find P(Y1>21₂ = 3). j) Are Y₁ and Y2 independent? Why? k) Find E(Y₁). 1) Find V(Y₁₂). m) Suppose that f(y₁, y2) = ke-6y1-3y2 for y2 > y₁> 0 and f(y₁, y2) = 0 elsewhere. Find the density function of Y₁. n) Given the join density function in the previous part, are Y₁ and Y2 independent? Why? 4. Let Y, and Y2 be jointly discrete random variables with probability function p₁, y2) = (3)³¹ (3)" where y₁ and y₂ are integers and 1 ≤ y ≤ y2, and p(y₁, y2) = 0 elsewhere. a) Find the marginal probability function of Y₁, i.e., P₁(y₁). b) Find the conditional probability function of Y2 given Y₁ = y₁, i.e., p(y2ly₁). c) Find Σall y2 y2P(Y2ly1).