Homework set 1, Stat 4158, Spring 2021 Due date: 2 pm, Tuesday, Jan. 19, 2021 Instructions: Answer the following questions, all of which are about expectation of random variables, discussed in...

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Homework set 1, Stat 4158, Spring 2021 Due date: 2 pm, Tuesday, Jan. 19, 2021 Instructions: Answer the following questions, all of which are about expectation of random variables, discussed in sections 3.3., 4.3, 5.5 and 5.6 in the textbook. Review those sections, which should have been covered in Stat 4157. Write formal proofs and derivations and explain your answers appropriately. Submit your work before Tuesday’s class. 1. Let X be a Poisson random variable with parameter λ. Derive E(X) and E(2X). 2. Let Y be a continuous random variable with pdf fθ(y) = αy α−1/θα, if 0 < y="">< θ="" 0,="" otherwise.="" where="" α="" is="" a="" given="" constant,="" but="" θ=""> 0 is an unknown parameter. (a) Derive Eθ(Y ) and variance of Y . (b) Find a function g(Y ) of Y such that Eθ[g(Y )] = θ for all θ > 0. (c) Let Z = 3Y − 2. Derive the expected value and variance of Z. 3. Let X and Y be two continuous random variables with joint pdf f(x, y) = 4xy, if 0 < x="">< 1,="" 0="">< y="">< 10, otherwise. (a) derive e(x) and variance of x. (b) verify directly that e(x2y ) = e(x2)e(y ). (c) derive cov(x, y ) = e[{x − e(x)}{y − e(y )}], the covariance between x and y . (d) derive the mean and variance of x + 2y . 10,="" otherwise.="" (a)="" derive="" e(x)="" and="" variance="" of="" x.="" (b)="" verify="" directly="" that="" e(x2y="" )="E(X2)E(Y" ).="" (c)="" derive="" cov(x,="" y="" )="E[{X" −="" e(x)}{y="" −="" e(y="" )}],="" the="" covariance="" between="" x="" and="" y="" .="" (d)="" derive="" the="" mean="" and="" variance="" of="" x="" +="" 2y="">