8. a.Now consider a dice game: no points for rolling a 1, 2, or 3; 5 points for a 4 or 5; 50 points for a 6. Let X be the points earned. Find the probability distribution function. Find the expected value and the standard deviation of X. b. Consider the same game above but imagine doubling the points awarded. Find the probability distribution function(make the table). What are the new mean and standard deviation? You can use direct calculations (making the probability table and using the definition of expected value and variance or you can use the short cut formulas) and the properties of expected values and variances. c. Now imagine playing the same game twice. Find the probability distribution function . What are the mean and the standard deviation of your total points? You can use direct calculations or properties of expected values and standard deviations. Notice that this is not the same situation as playing once for twice as much. Think about tossing a coin once heads you win $50. That's not the same as tossing a coin five times at $10 a head. d. Suppose you and a friend both play the dice game. Find the probability distribution function. What are the mean and standard deviation of the difference in your winnings? Use the direction method or use the properties of expected values and variances