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Chapter 7: Q. 7.50 (page 363)

LetX have moment generating function M(t), and defineΨ(t)=logM(t). Show thatΨ''(t)t=0=Var(X).

Short Answer

Expert verified

The second derivative value of Ψ(t)and plug int=0.

Step by step solution

01

Given Information

Let's Xhave moment generating functionM(t)show that Ψ''(t)t=0=Var(X)̣

02

Explanation 

The expression for the second derivative of Ψ(t)=logM(t),

Ψ'(t)=M'(t)M(t)

Which implies

Ψ''(t)=M''(t)M(t)-M'(t)M'(t)M2(t).

03

Explanation

Now, plug t=0into the previous expression to obtain that,

Ψ''(t)t=0=M''(0)M(0)-M'(0)M'(0)M2(0)

=EX2-E(X)2=Var(X)

so we have proved the claimed.

04

Final answer

The second derivative value of Ψ(t)and plug int=0.

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