3. A box contains 4 black balls and 6 white balls. A player randomly draws one ball and returns the ball to the box along with 2 additional balls with the same color. Then the player randomly draws another ball. Define the random variables Y₁ = { 1, if the first ball is black, 0, if the first ball is white, Y2 = { 1, if the second ball is black, 0, if the second ball is white. 1) Show that the joint probability mass function (PMF), f(y1, y2), of (Y₁, Y2) is given by Table for f(y1, Y2) Y1 0 Y2 1 0 1 2515 2) Find the marginal PMF, f2(y2), of Y2. 3) Find E(Y2) and var(Y2). 4) Find the conditional PMF, f21 (y2|y1), of Y2 given Y₁ = y₁.. 5) Find P(Y₁+Y₂ <1), the probability that the player obtains at most 1 black ball. 6) Find cov(Y1, Y2). Explain why the sign of cov (Y₁, Y2) you obtain is sensible. 7) Find E(Y1Y2) and var(Y₁ + Y2). 8) The player is awarded $1 if the two balls obtained are of the same color, and $2 if the two balls are of different colors. Let Z be the amount of dollars awarded to the player. Find E(Z) and var(Z). 9) Find E(Y2) and var(Y2) using the formulas E(Y2) = E[E(Y2Y1)] and var(Y2) = var[E(Y₂|Y₁)]+ E[var(Y2Y1)].