1 To purchase a certain high-level weapon in Final Fantasy XII, you must collect and sell specific kinds of loot (items that the monsters drop after battle or are stolen during battle). To acquire one of the best 2-handed swords in the game, called the Tournesol, you need an item called a Gemsteel; they can only be found by fighting a Level 99 Red Chocobo (a large bird somewhat similar to an ostrich). To attempt to spawn him requires doing a specific sequence of events to attempt to \"draw\" him outEven with extensive farming, the chocobo only has a 1/256 chance of spawning; although he does not show up that often, when he finally does, the gemsteel is an almost guaranteed drop (the gemsteel has a 90% drop rate, which is irrelevant for this problem). For now, farming runs will consist of 10 spawning attempts. Let X = the number of times the Level 99 Red Chocobo appears n = the number of tries to find the bird = 10 p = probability of success (the chance of the Level 99 Red Chocobo showing up) 1. If you are looking for the probability of k success out of n trials, what is the name of the probability distribution you would use? List its parameter(s) and the numerical value(s). 2. Calculate all of the individual probabilities (P(X = 0), P(X = 1), , P(X = 10) ) using the distribution you just chose 3. Calculate the mean, variance, and standard deviation of the Level 99 Red Chocobo's appearance in the 10 attempts. 4. Now suppose that I decide I want to do a farming run of 100 tries to spawn him (which would indicate I have no life. . . ). Calculate EX , V X , and SDX with n = 100 and p = 1 256 5. Calculate the probability that the chocobo will appear at least once (out of 100 tries) using R . Problem 2 A service station had both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time Let Y denote the number of hoses on the full-service island in use at that time The joint pmf of X and Y are in the following table. y p(x, y) 0 1 2 0 0.15 0.02 0.01 x 1 0.06 0.18 0.07 2 0.03 0.18 0.30 1. Calculate the Marginal distributions of X and Y 2. Calculate EX, EY, EXY, VX, VY, SDX, SDY Problem 3 The article \"Determination of the MTF of Positive Photoresists Using teh Monte Carlo Method\" (Photographic Sci. and Engr., 1983: 254-260) proposes the exponential distribution with parameter _ = 0.93 as a model for the distribution of a photon's free path length (m) under certain circumstances. Use this model with parameter _ = 0.93. f(x) = _e_x, x _ 0, 0 elsewhere 1. Calculate the expected path length, variance and standard deviation of path length 2. Calculate the probability that the path length exceeds 3.0 m 3. Calculate the probability the path length is between 1.0 and 3.0 m 4. Find the CDF For up to 5 points extra credit (turn in regardless of whether or not this problem is collected for exam credit): What value is exceeded by only 10% of all path lengths (the top 10%)? Show all work for partial credit (either by hand or in R) 3