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Question 4: Expectation and Variance (10 marks) Let X be a continuous random variable and Y= ox + b for some constants o and b prove, using the definition for expectation and variance, the following 1. E(Y ) = GE(X) + b. (5 marks) Y = ax + b E(n) =E( ar + b) ( ar + b) + (x) ar ar + ( x ) dx + ( b+ (x) dx za. * + (x) dx +b+ + (x) dx =of(x) + b 2. Var( Y ) = o'Var[X). (5 marks) Var( ) = Var( ar+ b) =E[(ar+b) ]] - E3(ar+b) = (ar + b) + (x) dx - [aE(x) + 6] ma x24 (x) dx + 2ab / x+ (x) dx + bi / + (x) dx - 43 82 (x) ma' E( x3) 2abE( x) + b3 - a3 E' (x) - 2abE(x) - bi 20' E(x) 2of (x) + 5 - 43 83 (x) - 30#f (x) - =0? [E(x3) - 83 (x)] mad var (x) Total: 10 marks