I have a problem that asks me to show that $Y_{(1)}$ is a biased estimator for $\theta$ where a random sample of size n is taken from a uniform distribution $U(0,\theta)$.
I am stuck on integrating the mean $E{(Y_{(1)})}$. I let $$g_1(y) = \frac n \theta[1-\frac Y \theta] ^{n-1}$$ be the density function of $Y_{(1)}$. Therefore I let $$E(Y_{(1)}) = \int_0^\theta \frac y n \theta (1-\frac y \theta)^{n-1} dy$$
I have no idea where to start with this integral. I know this is what is needed to be solved to show it is a biased estimator. Can someone help me break down this integral showing all steps?
Thank you
Just substitute $1-y/\theta = x$, then $$ dy = -\theta dx, $$ and you will get an immediate integral.