(Solved): A friend bets you \( \$ 10 \) that she will outscore you on the exam. Let \( \mathrm{X} \) be a ra ...

A friend bets you \( \$ 10 \) that she will outscore you on the exam. Let \( \mathrm{X} \) be a random variable denoting your winnings. \( X \) can take on the following values: \( 10,-10,0 \) ( 0 if you tie). You know that the outcome depends on whether she studied for the exam. Let \( Y=1 \) if she studied and 0 if she did not. The joint distribution is: a. Fill in the missing elements in the joint and marginal distributions, using the information that \( \operatorname{Pr}(\mathrm{X}=0 \mid \mathrm{Y}=0)=.05 \) b. Compute \( \mathrm{E}(\mathrm{X}), \mathrm{E}(\mathrm{Y}), \operatorname{Var}(\mathrm{X}) \) and \( \operatorname{Var}(\mathrm{Y}) \). Should you take the bet? Why or why not? c. What is the expected value of your winnings if your friend did study for the exam? If she did not study?