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With reference to Exercise 6 39 show that for nor mal distributions

With reference to Exercise 6.39, show that for nor–mal distributions k2= σ2 and all other cumulants are zero.

In exercise

If we let KX(t) = lnMX – µ(t), the coefficient of tr/r! in the Maclaurin’s series of KX(t) is called the rth cumulant, and it is denoted by kr. Equating coefficients of like powers, show that

(a) k2 = µ2;

(b) k3 = µ3;

(c) k4 = µ4 – 3µ22

In exercise

If we let KX(t) = lnMX – µ(t), the coefficient of tr/r! in the Maclaurin’s series of KX(t) is called the rth cumulant, and it is denoted by kr. Equating coefficients of like powers, show that

(a) k2 = µ2;

(b) k3 = µ3;

(c) k4 = µ4 – 3µ22

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