# The probability distribution function raised to a power is also a probability distribution function?

I got this tutorial question and I had no clue about how to prove this:

Let $\,F(x)\,$ be a distribution function and $\,r\,$ a positive integer.

Show that $\,F(x)^r\,$ is also a distribution function

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Hint: Which properties qualify a function $G:\mathbb R\to\mathbb R$ to be a distribution function? – Did Sep 23 '12 at 12:35
Thanks ! Its non-decreasing. F(-infinity) = 0 and F(infinity) = 1. These still hold when exponentiated if r is positive. This was pretty obvious - I was being stupid – user929404 Sep 23 '12 at 12:43
These are not sufficient to guarantee that F is a distribution function. There is still one more property... – Did Sep 23 '12 at 12:44
@did I remember similar story from another previous question) – Seyhmus Güngören Sep 23 '12 at 13:05
hope you were referring to continuous from the right – user929404 Sep 23 '12 at 18:20

Hint: Which properties qualify a function $G:\mathbb R\to\mathbb R$ to be a distribution function? – did

It's non-decreasing. $F(-\infty) = 0$ and $F(\infty) = 1$. These still hold when exponentiated if $r$ is positive. – user929404

These are not sufficient to guarantee that F is a distribution function. There is still one more property... – did

continuous from the right - user929404

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