CRLB for p: Find E[x] with Maclaurin Series

• neznam
In summary, the conversation is about finding the CRLB for a distribution with a given probability density function and parameter, and the difficulties in finding the expected value of the distribution. The conversation includes a suggestion to use a Maclaurin series to solve the problem. Eventually, the expected value is determined to be lnp, and the conversation ends with a thank you.

neznam

Random Sampe of size n from distribution with pdf
f(x;p)={(lnp)^x}/px! for x=0,1,...; p>1 and 0 otherwise

Find CRLB for p?

My problem is finding E[x] which is somekind of maclaurin series but can't figure out which one?

Thanks

I'm not familiar with the abbreviation CRLB. What does it mean?

What expression do you have for E[x]? You need to show more work for us to see where you're getting stuck.

E[x] = [1*(lnp)^1]/p*1!+[2*(lnp)^2]/p*2!+[3*(lnp)^3]/p*3!+...

Once I have the expected value E[X] of this distribution I will be able to find the CRLB as well which is defined to be in this case

1/(n*[d/dp ln f(x;p)]^2

Any help is appreciated

So my main problem is figuring out the E[X] and as a hint of this problem it is saying to use Maclaurin series.

If you factor a 1/p out from your original f then you got z^k/k! (where I'm writing z=ln(p) and k=x to make it look more like a maclaurin series. Do you recognize what function that is? The expectation value is then associated with the maclaurin series k*z^k/k!=z^k/(k-1)!. Can you modify the function you recognized to get the second series?

sum of z^k/k! is e^k, so is it sum of z^k/(k-1)! will be e^(k-1)?

neznam said:
sum of z^k/k! is e^k, so is it sum of z^k/(k-1)! will be e^(k-1)?

Sum of z^k/k! is e^z. Try that again.

Ok i think i got it

E[x]=[(lnp)/p]*e(lnp)=lnp

Thanks so much

What is the CRLB for p?

The Cramér-Rao lower bound (CRLB) for p is a theoretical lower limit on the variance of unbiased estimators of a parameter p in a statistical model. It provides a benchmark for the best possible performance of unbiased estimators and can be used to evaluate the efficiency of different estimation methods.

How is the CRLB for p calculated?

The CRLB for p is typically calculated using the Fisher information, which is a measure of the amount of information that a sample of data provides about the parameter p. The CRLB is then derived by taking the inverse of the Fisher information matrix.

Why is the CRLB important?

The CRLB is important because it helps to determine the minimum achievable variance for an unbiased estimator of a parameter. This can be used to evaluate the efficiency of different estimation methods and to compare the performance of different estimators.

What is the Maclaurin series?

The Maclaurin series is a special case of the Taylor series, which is a mathematical representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. In the Maclaurin series, the single point is chosen to be 0.

How can the Maclaurin series be used to find E[x]?

The Maclaurin series can be used to find the expected value (E[x]) of a function by taking the first derivative of the function at 0 and multiplying it by x, and then taking the second derivative at 0 and dividing it by 2 factorial and multiplying it by x^2, and so on. The sum of these terms will give the expected value of the function at x=0.