Question 1 Let X1, X2 be a random vector with the joint PDF 2 0 <...

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Question 1 Let X1, X2 be a random vector with the joint PDF 2 0 < x1 < X2 < 1 f(x1,x2) otherwise a) Draw the region where f(x1, x2) > 0 b) Find the pdf of the marginal distribution of X. c) Find the conditional distribution of X2 given X = a for some a (0, 1) by deriving its conditional pdf. d) Find the joint density function of (- log(X1), - log(X2)). Question 3 300 people attempt a difficult challenge, each person's attempt is indepen- dent, and they all have a 0.15 chance of success. Let X be the total number of successes in this group of 300 people. a) What is the distribution of X? Give parameters. Obtain the proba- bility of P(X90) to 5 significant digits using R or other statistical computing software (present value in scientific notation if it gets too small). b) Give an upper bound for P(X > 90) using Markov's inequality. c) Give an upper bound for P(X > 90) using Chebychev's inequality. d) Give an approximation of P(X >90) based on the CLT. e) Do the results regarding P(X > 90) from parts a to e contradict each other? Why or why not? Question 1 Let X1, X2 be a random vector with the joint PDF 2 0 < x1 < X2 < 1 f(x1,x2) otherwise a) Draw the region where f(x1, x2) > 0 b) Find the pdf of the marginal distribution of X. c) Find the conditional distribution of X2 given X = a for some a (0, 1) by deriving its conditional pdf. d) Find the joint density function of (- log(X1), - log(X2)). Question 3 300 people attempt a difficult challenge, each person's attempt is indepen- dent, and they all have a 0.15 chance of success. Let X be the total number of successes in this group of 300 people. a) What is the distribution of X? Give parameters. Obtain the proba- bility of P(X90) to 5 significant digits using R or other statistical computing software (present value in scientific notation if it gets too small). b) Give an upper bound for P(X > 90) using Markov's inequality. c) Give an upper bound for P(X > 90) using Chebychev's inequality. d) Give an approximation of P(X >90) based on the CLT. e) Do the results regarding P(X > 90) from parts a to e contradict each other? Why or why not?