6. Let W = Yu, where Y~n(μ, o²). Find the moment-generating function of W. 1

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6. Let W = Yu, where Y~n(u, o2). Find the moment-generating function of W.
The Moment-generating Function Method of finding a sum of variables.
PROPERTY D
If X and Y are independent random variables with mgf's Mx and My and Z =
X + Y, then Mz(t) = Mx (1)My (1) on the common interval where both mgf's
exist.
Proof
Mz(t) = E(e¹z)
= E(e¹X+¹Y)
= E(e¹xe¹)
1
From the assumption of independence,
Mz(t) = E(e¹x)E(e¹)
= Mx (1) My (1)
1/2
Transcribed Image Text:6. Let W = Yu, where Y~n(u, o2). Find the moment-generating function of W. The Moment-generating Function Method of finding a sum of variables. PROPERTY D If X and Y are independent random variables with mgf's Mx and My and Z = X + Y, then Mz(t) = Mx (1)My (1) on the common interval where both mgf's exist. Proof Mz(t) = E(e¹z) = E(e¹X+¹Y) = E(e¹xe¹) 1 From the assumption of independence, Mz(t) = E(e¹x)E(e¹) = Mx (1) My (1) 1/2
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