1. Suppose that Z1,Z2 are independent standard normal random variables. Let Y1 = Z1 − 2Z2,...

1. Suppose that Z1,Z2 are independent standard normal random variables. Let Y1 = Z1 − 2Z2, Y2 = Z1 − Z2.

(a) Find the joint pdf fY1,Y2(y1,y2). Don’t use the change of variables theorem – all of that work has already been done for you. Instead, evaluate the matrices Σ and Σ−1, then multiply the necessary matrices and vectors to obtain a formula for fY1,Y2(y1,y2) containing no matrices and no vectors.

(b) Find the marginal pdf fY2 (y2). Don’t use integration.

(c) Find the conditional pdf of Y1 given Y2 = 1, that is, fY1|Y2(y1|1). Use (a) and (b), and do the necessary division. By completing the square, identify by name – including pa- rameters – the required conditional pdf.

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orchestra answered 1 year ago

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