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LINEAR TRANSFORMATIONS OF NORMAL RANDOM VARIABLES ARE ALSO NORMAL

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Author: Christine Farr
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STAT 400G. FellourisFall 2015Homework 8Readings• Sections 3.3 (Normal distribution), See also Tables Va and Vb in pg. 494-495.• Section 4.1 (Joint distribution and Independence)• Section 4.3 (Conditional distributions)Exercises1. Linear Transformations of Normal Random Variables are also Normal(a) Let X ∼ N (µ, σ 2 ) and a, b ∈ R. Show that Y = aX + b is also a normal randomvariable. With what parameters?Hint: Compute FY (y) = P(Y ≤ y) = P(X ≤?) =. . ..(b) Suppose that the temperature in some town on some randomly chosen day followsthe normal distribution with mean 15 and standard deviation 5, when measuredin o C. What can you say about the temperature in o F ?Hint: o F = 32 + (9/5) o C.2. Moment-generating function of the normal(a) Let Z ∼ N (0, 1) and show thatMZ (t) = E[exp{tZ}] = expt22,t ∈ R.Hint: This formula is obtained with a technique called ”completion of a square”,which is often used with Gaussian random variables. See pg. 106.(b) Let X ∼ N (µ, σ 2 ) and show thatMX (t) = E[exp{tX}] = exp µt +σ2 2t ,2t ∈ R.Hint: Use (a) and the fact that X can be expressed as X = µ + σZ, whereZ ∼ N (0, 1).

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