# If $X$ is uniform on the interval $[0, 2]$ , find the PDF of $X^2-2X$

If $X$ is uniform on the interval $[0, 2]$ find the PDF of $Y = X^2-2X$.

I solved for $X$ and got $x=1 + \sqrt{1+y}$ or $x=1 - \sqrt{1+y}$. Not sure what to do from here.

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This is likely a homework, please tag it appropriately if so. You can get an answer using Mathematica as well, try PDF[TransformedDistribution[x^2 - 2 x, Distributed[x, UniformDistribution[{0, 2}]]], x] –  Sasha Jan 17 '13 at 16:11
Thanks! Yes it is, and I do not have mathematica. –  Olivia Irving Jan 17 '13 at 16:23

## 1 Answer

Note that $Y=(X-1)^2-1$. Now let $W=X-1$. It is nearly obvious that $W$ is uniform on $[-1,1]$.

We find the distribution of $W^2$. Since $Y=W^2-1$, it is then easy to find the distribution of $Y$.

Let $Z=W^2$. Then (for $0\le z\le 1$), $$\Pr(Z\le z)=\Pr(W^2\le z)=\Pr(|W|\le \sqrt{z})=\frac{2\sqrt{z}}{2}=\sqrt{z}.$$

Remark: The problem can also be solved using a variant of your calculation. We have $Y\le y$ if and only if $X^2-2X\le y$. This inequality holds if and only if $$1-\sqrt{1+y}\le X\le 1+\sqrt{1+y}.$$ If $y\ge 0$, the above probability is $1$. So the cdf of $Y$ is $1$ for $y \ge 0$.

If $y\lt -1$, the above condition is an impossible one, so the probability is $0$. Thus the cdf of $Y$ is $0$ when $y\lt -1$. So the only interesting $y$ are between $-1$ and $0$.

For $-1\le y\le 0$, the length of the interval $[1-\sqrt{1+y},1+\sqrt{1+y}]$ is $2\sqrt{1+y}$, and the interval lies within $[0,2]$. So the probability we land in that interval is $\frac{2\sqrt{1+y}}{2}$, that is, $\sqrt{1+y}$.

Now that we have the cdf of $Y$, we can differentiate to get the density.

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Where did the Y=W^2-1 go? I understand Z=W^2 but I am not sure where the -1 went. –  Olivia Irving Jan 17 '13 at 16:28
I showed how to find the cumulative distribution function for $W^2$. It was left to you to use this to find the cdf of $W^2-1$. –  André Nicolas Jan 17 '13 at 16:32