I am having troubles with 1 (d) and could use some direction. Would the expected value of x be equal to the expected value of y and would the variance of x we equal to the variance of y also?

Homework #3 For a problem that asks you to use R, include a copy of the code and output. 1. A nefarious gambler has developed a weighted six-sided die, with sides marked 1, 2, 3, 4, 5, and 6. The probability that a 6 is rolled is 0.4, and all the other sides are equally likely. Define the random variable X as the value on the die when it is rolled once. (a) (b) (c) (d) Write the pmf of X. That is, make a table of the values X can take and the associated probabilities. Find the probability that an even number is rolled. Find the mean (or expected value) and variance of X. The gambler has a second weighted die designed like the first. A game is based on the number W = 2X + Y , where X is the number that shows up on the first weighed die and Y is the number that shows up on the second. i. Find the expected value (or mean) of W . ii. Find the standard deviation of W . 2. A high school student applies to seven colleges, each of which makes an admission decision by rolling a fair six-sided die and admitting a student if a 1 is rolled. What is the probability the student will be admitted to exactly two colleges? (Round your answer to two decimal places.) 3. To celebrate their 30th birthdays, brothers Mario and Luigi of the Nintendo Mario video game franchise wish to study the distribution of heights of their mushroom enemies, the Goombas. Their reasoning is that shorter Goombas are easier to jump on. (Goombas die when Mario and Luigi jump on them.) (a) If we assume that the population of Goomba heights are normally distributed with mean 12 inches and standard deviation 6 inches, what is the probability that a randomly chosen goomba has a height between 13 and 15 inches? (b) Koopa Troopas, other enemies of Mario & Luigi, have a mean height of 15 inches with standard deviation 3 inches. What is the probability that a randomly chosen Koopa Troopa is taller than 75% of Goombas? 4. The Hereford Cattle Society says that the mean weight of a one-year-old Hereford bull is 1135 pounds, with a standard deviation of 97 pounds. Suppose 40 bulls are randomly selected and loaded on a train car. Find the probability their combined weight exceeds 46000 pounds. (Hint: The combined weight exceeds 46000 pounds if the average weight exceeds 46000 40 = 1150 pounds.) 5. Let F be an RV that represents the operating temperature in Fahrenheit of one instance of a manufacturing process, and let F N (90, 52 ). Let C be an RV that represents the same process, but measured in Celsius. Fahrenheit can be converted to Celsius using C = 95 (F 32). (I recommend doing these with a calculator and N (0, 1) table as practice for the exam. Then check your answers with R if you wish.) (a) Find the probability that one randomly selected instance of the process will have operating temperature greater than 93.8 Fahrenheit. (b) C is also normally distributed. Find its mean and variance. (c) Find the probability that one randomly selected instance of the process will have operating temperature below 29 Celsius. (d) Find the Celsius temperature x such that the probability that the operating temperature in Celsius of one instance is less than x is .25. 1