Mean of a continuous random variable is given by :
where density function is given by
My question is how can I calculate the mean of a continuous random variable when the density function takes the mean itself as a parameter. And that density function is required to calculate the value of the mean.
You are supposed to start with the formula for the density and prove that $\mu$ is indeed the mean. You are not supposed to assume that $f$ here stands for a pdf with mean $\mu$ and variance $\sigma ^{2}$.
Here is the proof: $\int xf(x) dx=\int (x-\mu) f(x)dx+ \mu \int f(x) dx$. The first term is $\int yf(y+\mu) dy$ by the change of variable $y=x-\mu$. This inetgral is $0$ because the integrand is an odd function. Hence we are left with $0+(\mu )1=\mu$. I have only used the fact that $f$ is a density function.