Transformation of random variables (square, root)

Question

Let $X$ ~ Unif$([0,10])$ be a continuous random variable. I now want to find the probability mass function of $Y :=- \frac{1}{2}X+4$. I did some reading and figured out that $Y$ ~ Unif$([-1,4])$ with $f_{Y}(y)=f_{X}(-2y+8)\cdot 2 \cdot 1_{[-1,4]}$. So far, so good.

However, I have no idea how to find the probability mass function and distribution function of $\sqrt{X}$ and $X^{2}$. I would be grateful for any tip into the right direction. Does it work the same way as with linear transformations of random variables?

Answer 1

Hint:

Find CDF's of $\sqrt X$ and $X^2$.

Then find PDF's (not PMF as you call it) by differentiating the CDF's.

Answer 2

Transformation works the same way, linear or not.

If $Y=g(X)$, then $x=g^{-1}(y)$ if unique inverse exists (as in both these cases). So you have a one-to-one transformation from $X$ to $Y$. (This would not be the case in general)

Find the support $S^*$ of $Y$ from the support of $X$.

Then by change of variables, probability density function of $Y$ is

$$f_Y(y)=f_X(g^{-1}(y))\left|\frac{d}{dy}g^{-1}(y)\right|\mathbf1_{y\in S^*}$$

For linear transformations, the jacobian $\frac{d}{dy}g^{-1}(y)$ is constant. Here that is not the case.