Suppose X₁ and X₂ have the following joint distribution X₁ 1 d 2 1/3 1/12 1/6 0 2 1/36 0 0 0 X₂ 1 0 1/6 2/9 Find the covariance of X₁ and X₂. Are X₁ and X₂ independent? Compute the variance of Y = 2X₁ - 3X₂ + 1. Compute the correlation of Y and X₁.

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Arrange these steps in the correct order to prove that if you have an 8-gallon jug of water and two empty jugs with capacities of 5 gallons and 3 gallons, respectively, then you can measure 4 gallons by successively pouring some of or all of the water in a jug into another jug in the correct order. Rank the options below. Fill the 3-gallon jug from the 5-gallon jug, leaving (3, 2, 3). Empty the 5-gallon jug's contents into the 3-gallon jug, leaving (6, 0, 2), and then fill the 5-gallon jug from the 8-gallon jug, producing (1, 5, 2). Fill the 5-gallon jug from the 8-gallon jug, leaving the contents (3, 5, 0), where the ordered triple is used to record the amount of water in the 8-gallon jug, the 5-gallon jug, and the 3-gallon jug, respectively. Top off the 3-gallon jug from the 5-gallon jug, and there will be (1, 4, 3), with four gallons in the 5-gallon jug. Pour the contents of the 3-gallon jug back into the 8-gallon jug, leaving (6, 2, 0). ❤ ← ←