Hypothesis test on transformed data

[A_B]

Say I have a sample A with mean μ, and the log transformation of A, lnA. Is there any way of figuring out the mean of lnA? what if the distribution of lnA is normal?

thanks
Alex

 

[chiro]

Say I have a sample A with mean μ, and the log transformation of A, lnA. Is there any way of figuring out the mean of lnA? what if the distribution of lnA is normal?

thanks
Alex

Hello A_B and welcome to the forums.

In this problem you just use the definition of E[h(X)] where h is a function and X is your random variable.

Basically you just have to use the formula that E[h(X)] = Integral (over some domain) h(x) pdf(x) dx for a continuous variable.

The key thing though is because you are using a log function, you need to adjust your domain to suit that. If for example your distribution A was normal, then you couldn't apply your transform over the whole of A (since the domain of a normal is the whole real line). So just be careful when you're defining the range so that ln(x) is valid for this domain.

If your distribution is discrete then instead of an integral, replace that with a summation. If you're confused about what I'm saying grab any introduction statistics book and look at the definition of expectation.

Also if ln(A) was normally distributed then A >= 0 for its domain, so that's something easy to check. One way to get the pdf of A if ln(A) was normal is to use transformation rules with PDF's. Since ln(A) has an inverse transformation (e^(x)), you should be able to use the transformation to get firstly a CDF and then a PDF (differentiating).