Hello Nathalie,
To calculate the mean of a function θ over an interval 
with respect to a cumulative distribution function (CDF) 
, you need to evaluate the integral:
with respect to a cumulative distribution function (CDF) , you need to evaluate the integral:
This is essentially the expectation of 
over the interval 
. If 
is a CDF, then its derivative, 
, is the probability density function (PDF). The integral can be rewritten using this PDF:
over the interval . If is a CDF, then its derivative, , is the probability density function (PDF). The integral can be rewritten using this PDF:
Steps to Calculate the Mean:
- Identify the PDF: Differentiate the CDF
to obtain the PDF 
. - Set Up the Integral: Use the PDF to set up the integral for the mean:
- Evaluate the Integral: Compute the integral using analytical or numerical methods.
I hope this helps!
