Problem 17 Let $\psi(t)=\log M(t)$, where... [FREE SOLUTION]

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Introduction to Mathematical Statistics

Robert V. Hogg, Joeseph McKean, Allen T Craig

$Math Studyset 91影视 Explanations$ Math

7 Edition

Chapter 1: Problem 17

Let $\psi(t)=\log M(t)$, where $M(t)$ is the mgf of a distribution. Prove that $\psi^{\prime}(0)=\mu$ and $\psi^{\prime \prime}(0)=\sigma^{2} .$ The function $\psi(t)$ is called the cumulant generating function.

Short Answer

Expert verified

The first derivative of $\psi(t)$ at $t = 0$ is $\mu$ and its second derivative at $t = 0$ is $\sigma^2$.

Step by step solution

Calculate the First Derivative of $\psi(t)$

Firstly, apply the chain rule to find the first derivative of $\psi(t) = \log[M(t)]$. The derivative of $\log[x]$ is $1/x$, so using the chain rule the first derivative of $\psi(t)$ is $\psi'(t) = (1/M(t)) * M'(t)$ or $\psi'(t) = M'(t) / M(t)$.

Find the Value of $\psi'(0)$

Now, evaluate $\psi'(t)$ at $t = 0$. By the properties of the moment generating function, $M(0) = 1$ and $M'(0) = \mu$. Therefore, $\psi'(0) = \mu / 1 = \mu$.

Calculate the Second Derivative of $\psi(t)$

The second derivative of $\psi(t)$ requires differentiating $\psi'(t) = M'(t) / M(t)$ . By using the quotient rule which states that the derivative of $u/v$ is $(vu' - uv') / v^2$, we get $\psi''(t) = (M(t)*M''(t) - [M'(t)]^2) / [M(t)]^2$.

Find the Value of $\psi''(0)$

Evaluate $\psi''(t)$ at $t = 0$. By the properties of the moment generating function, we know already that $M(0) = 1$, and $M''(0) = \sigma^2$. Lastly, $[M'(0)]^2 = \mu^2$. Substituting these values, we find that $\psi''(0) = [\sigma^2 - \mu^2] / [1]^2 = \sigma^2 - \mu^2$. Since this is equivalent to the variance $\sigma^2$, we have proved our statement.

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91影视

Let \(\psi(t)=\log M(t)\), where \(M(t)\) is the mgf of a distribution. Prove that \(\psi^{\prime}(0)=\mu\) and \(\psi^{\prime \prime}(0)=\sigma^{2} .\) The function \(\psi(t)\) is called the cumulant generating function.

Short Answer

Step by step solution

Calculate the First Derivative of \(\psi(t)\)

Find the Value of \(\psi'(0)\)

Calculate the Second Derivative of \(\psi(t)\)

Find the Value of \(\psi''(0)\)

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