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The followings are the results of the first derivatives with respect to $\mu$ and $\sigma^2$

$\dfrac{\partial \ln l}{\partial\mu} = \dfrac{1}{\sigma^2}\sum_{i=1}^n(x_i-\mu)$

and

$\dfrac{\partial \ln l}{\partial\sigma^2} = -\dfrac{N}{2\sigma^2} + \dfrac{1}{2\sigma^4}\sum_{i=1}^n(x_i-\mu)^2$

The result of the second derivative with respect to $\mu$ is as follows:

$\dfrac{\partial^2 \ln l}{\partial\mu\partial\mu} = \dfrac{\partial \ln l}{\partial\mu^2} = -\dfrac{N}{\sigma^2}$.

However, I wonder the algebraic steps to get the result.

and the second derivative with respect to $\sigma^2$ is defined as follows:

$\dfrac{\partial^2 \ln l}{\partial(\sigma^2)^2}$.

I also wonder the algebraic steps to get the result for this, too.

user122358
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1 Answers1

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Check your first equation; I don't think it's consistent with your second equation. It doesn't alter the nature of my guidance toward an answer though.

Differentiating the first equation with respect to $\mu$: split the sum up, write it in the form: $\text{constant} - \text{constant} \times \mu$ and apply basic rules of differentiation.

Differentiating the second equation with respect to $\sigma^2$: write it as $\text{constant}\times (\sigma^2)^{-1} + \text{constant} \times (\sigma^2)^{-2}$ and (treating $\sigma^2$ as if it were just a symbol in its own right), apply basic rules of differentiation of powers of the variable.

Differentiating the first equation with respect to $\sigma^2$: See it as $\text{constant}\times (\sigma^2)^{-1}$ and again apply basic rules of differentiation of powers of the variable.

Figuring out what the constants are is just straight algebraic manipulation.

Glen_b
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