Calculate mean of a CDF

Can you please help to calculate mean on some interval [a,b]: $\underset{a}{\overset{b}{\int}}\theta dF\left(\theta\right)$ knowing F(theta). Thanks in advance

Answer 1

[Prateekshya]

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:

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:

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!