# question about a proof of distribution

Hi all I have a question about a proof that I don't understand,

My question is about the line after "We also have that....", I don't understand how $$P(\hat{\theta_n} \geq \theta -\frac{x}{n})$$ becomes $$1-(1-\frac{x}{\theta*n})^n$$

Could someone kindly explains?

• Isn’t that simply $P(\hat{\theta}_n \geq ...) = 1 - P(\hat{\theta}_n \leq ...)$ and for the latter expression one used the equation above...? Dec 14, 2019 at 5:22
• (and then pull out the $\theta$ in $\theta-x/n$...) Dec 14, 2019 at 5:24
• please add the self-study tag and provide more details Dec 14, 2019 at 9:33

The result follows since \begin{align*}P(\hat\theta_n\geq\theta-x/n)&=1-P(\hat\theta_n<\theta-x/n)\\&=1-P(\hat\theta_n\leq\theta-x/n)\\&=1-\Big(\frac{\theta-x/n}{\theta}\Big)^n\\&=1-(1-x/(\theta n))^n,\end{align*} where the second equality follows by continuity of $$\theta_n$$ (it is continuous because its cdf is continuous), and where the third equality follows from the fact that since $$x\in[0,\theta]$$, $$(\theta-x/n)/\theta\in[1-1/n,1]$$, assuming $$\theta\geq1$$.