Find unbiased for $\theta$ of $\hat{\theta_1} = (n+1) ~y_{(1)}$given a uniform distribution on the interval $[0, \theta]$.

Question

Show that $\hat{\theta_1} = (n+1) ~y_{(1)}$ is unbiased for $\theta$.

For $$P[Y_i \le Y] = 1 - y/\theta$$ Then for $ P[Y_{(1)} < y] = 1 - [1-F_{(Y_i)} (y)]^n$ which should equal to $1 - [1-(1-y/\theta)]^n = 1 - [y/\theta]^n$ However, $F_{Y_{(1)}}(y)$ is supposed to be $1 - (1-y/\theta)^n$....where is my derivation goes wrong?

Answer 1

I think your whole class has posted this at some point in the last week or so. If $P[Y\le y]=1-y/\theta$ then $P[Y \le 0]=1$ and $P[Y \le \theta]=0$ which does not look right!