Natural logarithm of x

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May I ask how to find the [tex]E\left(ln x\right)[/tex] and [tex]Var\left(ln x)[/tex]?
The [tex]X_{i}[/tex] are random sample from the [tex]f\left(x; \theta\right) = \theta x^{\theta - 1}I_{\left(0, 1\right)}\left(x\right)[/tex] where [tex]\theta > 0[/tex].

I need the information in finally solving the Cramer-Rao lower bound for the variance of an unbiased estimator of a function of [tex]\theta[/tex]. And also for checking if I have an unbiased estimator.
 

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  • #2
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May I ask how to find the [tex]E\left(ln x\right)[/tex] and [tex]Var\left(ln x)[/tex]?
A random variable X has a lognormal distribution if [tex]X=e^{Y}[/tex] where Y is normally distributed with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex].

Then: [tex] E(X)=e^{\mu+\sigma/2}, Var(X)=e^{2(\mu+\sigma)}-e^{2\mu+\sigma}[/tex]

The moment generating function is: [tex]E(X^n)=e^{n\mu+(n^2\sigma^2/2)}[/tex]

Note the moments of the lognormal distribution do not uniquely define the distribution and a finite mgf only exists on the interval [tex](-\infty,0][/tex]

EDIT:: You can assign other distributions to Y, but without specification, I don't think there's a single answer to your question. The lognormal is the default under the Central Limit Theorem..
 
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  • #3
EnumaElish
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Let g(x)= ln x. For E[g(x)] see the inner product definition here. For Var[g(x)], look under continuous case here with g(x) replacing x and [itex]\mu[/itex] is interpreted as E[g(x)]; alternatively see the delta method.
 
  • #4
EnumaElish
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That is, Var[g(x)] = integral of ( g(x) - E[g(x)] )^2 f(x), over x's domain.

Alternatively, [tex]

Var[g(x)] = E\left[g(x)^2\right] - \left(E[g(x)]\right)^2
= \int{g(x)^2 f(x) dx} - \left(\int {g(x)f(x)dx}\right)^2

[/tex]

where each integral is over the domain of x.
 
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