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Let X1 and X2 be independent random variables having the standard normal distribution. Obtain the joint Lebesgue density of (Y1, Yz), where Yi = VX?+ X} and Y2 = X1/X2. Are Y1 and Y2 independent? Note. For this type of problem, we may apply the following result. Let X be a random k-vector with a Lebesgue density fx and let Y = g(X), where g is a Borel function from (R*, B* ) to (R*, B*). Let A1, . .., Am, be disjoint sets in B* such that R* - (Aj U . . . U Am,) has Lebesgue measure 0 and g on A; is one-to-one with a nonvanishing Jacobian, i.e., the determinant Det(Og(x)/Or) # 0 on Aj, j =1. .... m. Then Y has the following Lebesgue density: fy (x) = >Det (Oh, (x)/ar) | fx (h,(r)). j=1 where h, is the inverse function of g on A;, j = 1, ..., m.2. Suppose you ask a person at many random times to state how happy they are [on some scale}. These scores fora singie person are then averaged to get an average happiness snore [AHIL If you nd the AH score for different people. you'll curiously get different results, but let's suppose that the AH scores for random people are normaily distributed with mean 65 [on some particular happiness instrument}. a. Suppose you are told that the happiest 1% of people score 83 or above. What is the standard deviation of the AH distribution using this happiness instrument? b. A new happiness instrument is designed where AH is normallyI distributed with mean 50 and standard deviation 5. How marryr random peopie would you have to talk to before you met a total ofthree people who scored below 43 on this new instrument