84. TeamsI mage are all scheduled to play each of the other teams 10 times. Whenever team i plays team j, team i is the winner with probabilityI mage, where

(a) Approximate the probability that team 1 wins at least 20 games.
Suppose now that we want to approximate the probability that team 2 wins at least as many games as does team 1.
To do so, let X be the number of games that team 2 wins against team 1, let Y be the total number of games that team 2 wins against teams 3 and 4, and let Z be the total number of games that team 1 wins against teams 3 and 4.

(b) Are Image independent.

(c) Express the event that team 2 wins at least as many games as does team 1 in terms of the random variables Image.

(d) Approximate the probability that team 2 wins at least as many games as does team 1.
85. The standard deviation of a random variable is the positive square root of its variance. LettingI mage andI mage denote the standard deviations of the random variables X and Y, we define the correlation of X and Y by Image

(a) Starting with the inequality Image, show that Image.

(b) Prove the inequality

(c) If Image is the standard deviation of Image, show that Image