Z=(-1/sqrt(n)) * sum from k=1 to n of [1+log(1-Fk)] where Fk is a cumulative distribution function which is continious and strictly increasing.
Show that as n->infinity, Z converges to a normal distribution with mean 0 and var 1
Hm I don't understand. You mean it's the cdf taken at the value of a random variable, Xk ?
Isn't it that Fk is the cdf of Xk ?
And what kind of random variable is Xk ?
Why do you say Fk is uniform ? Would it mean that it's the cdf of a uniform distribution ? Over what interval ?
Is there the exact writing of the problem somewhere ?
Xk is an independent random variable and Fk is it's associated cumulative distribution function.
Sorry if I didn't make this clear. Missed this from the original question description.
So, Fk(Xk) will become a uniform distribution from 0 to 1 since it has equL likelihood of any vale from 0 to 1 given that Xk is independent and random. At least this is what I'm thinking now
Yes sorry that's right, given that Fk is strictly increasing (quite important assumption), $\displaystyle U_k=F_k(X_k)$ follows a uniform distribution over (0,1)
And we know (we can check it with cdf) that $\displaystyle -\log(1-U_k)$ follows an exponential distribution with parameter 1. And its expectation is 1.
So the expectation of $\displaystyle -(1+\log(1-U_k))$ is 0.
And here, we have $\displaystyle -\frac{1}{\sqrt{n}}\sum_{k=1}^n 1+\log(1-U_k)=\frac{1}{\sqrt{n}}\sum_{k=1}^n -(1+\log(1-U_k))$... so, can you apply the CLT here ?
This is good. Thanks.
I took the approach of calculating the moments of a log uniform by taking the standard approach but replacing the function, x, with ln(x) ie
1/(b-a) integral from a to b of ln(x)
Using this gave me -1 for the expected value and 1 for the variance.
I didnt think of using the quartile function and this looks to be a better approach. How do you calculate the variance though? Understand the mean is 1/parameter = 1
Find the variance of the exponential distribution of parameter 1 (that is 1 ;D) and since the random variables are independent, the variance of the sum is the sum of the variances !
The cdf is sometimes much more easier to use than calculating all the moments ^^ (use the inversion method)