4 - People.stat.sfu.ca
4 - People.stat.sfu.ca
4 - People.stat.sfu.ca
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The first 4 derivatives are<br />
M ′ X(t) = λ exp(t)M X (t)<br />
M ′′<br />
X (t) = M ′ X (t) + λ exp(t)M ′ X (t)<br />
M (3)<br />
′′<br />
X<br />
(t) = M X (t) + λ exp(t)M X ′ ′′<br />
(t) + λ exp(t)M X (t)<br />
M (4)<br />
(3)<br />
X<br />
(t) = M<br />
X (t) + λ exp(t)M X ′<br />
(t) + λ exp(t)M<br />
′′<br />
X<br />
Now plug in t = 0 in each formula to find:<br />
M ′ X (0) = µ X = λ<br />
M ′′<br />
X (0) = E(X 2 ) = λ + λ 2<br />
M (3)<br />
X (0) = µ′ 3 = λ + λ2 + λ 2 + λ(λ + λ 2 ) = λ + 3λ 2 + λ 3<br />
′′<br />
(3)<br />
(t) + λ exp(t)M X (t) + λ exp(t)M<br />
X (t)<br />
M (4)<br />
X (0) = µ′ 4 = λ + 3λ2 + λ 3 + λ 2 + λ(λ + λ 2 ) + λ(λ + λ 2 ) + λ(λ + 3λ 2 + λ 3 )<br />
= λ + 7λ 2 + 6λ 3 + λ 4 6
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